It was certainly not any kind of fringe opinion to suggest there would be a civil war. A better characterization is that, throughout the 1860 Presidential campaign, there was one political faction (the pro-secession Southern Democrats known as "Fire-Eaters") who were threatening and even promising secession, by forcible means if necessary, should Abraham Lincoln win the Presidency. They would wage war, if necessary, to secede from a Lincoln-led United States. The reaction among the other three factions was to play the blame game in various ways: if war came, was it the Southern Democrats' fault for taking up arms? Was it the Republicans' fault for meeting force with force? Was it one side or the other's fault—or everyone's fault—for not reigning in the radical elements in American politics, or for failing to compromise, or for failing to further appease the South?

Accusations and threats were thrown around regularly in the press, as well as in the legislative and political arenas throughout 1860 and early 1861. A couple of good compilations on this rhetoric can be found in the books Northern Editorials on Secession, ed. by Howard Cecil Perkins, and Southern Editorials on Secession, ed. by Dwight Lowell Dumond. For ease of reference, many of the editorials compiled in these books have been transcribed on the American Historical Association's website.

To give a sampling of the rhetoric that was being expressed throughout the 1860 election campaign and the ensuing Secession Crisis:

The Richmond Whig and Public Adverftiser published an editorial on March 19, 1860, more than a year before the war began, which accused the Southern Democrats of inciting war:

"... these Southern Democrats are ready to break up the Union, and plunge the country into civil war, on account of the North stealing a negro..."

Two months later, on May 26, 1860, the Louisville Daily Courier accused the Republican Party as being the instigators:

"The major wing of the Opposition [i.e., the Republicans], struggling for power, with the control of the Government almost in its reach, catering to the passions and to the prejudices of the ignorant, feeding the fires of fanaticism and madness, with the Constitution, fraternity, and equality on their lips, are pressing on to the extinguishment of slavery, though to reach this end it be necessary to trample under foot the Constitution, to invade the sovereignty of the States, to violate the rights of individuals, to dissolve the Union, to inaugurate strife, anarchy, and civil war."

The rhetroic was similar in the North. In a January 9, 1861, editorial, the New York Daily News blamed abolitionists and the Republican Party's unwillingness to compromise as the reason there might be war:

"...Abolition interests depend on war. Hence, 'no compromise.' Peace would now be more fatal than ever to the preachers of so called liberty, and they appeal to the God of War, to extreme issues, to the last argument of madmen."

In a February 5, 1861, editorial, the Buffalo Morning Express put the blame on the Southern secessionists' shoulders:

"The present attitude of affairs in South Carolina has a warlike tendency at the North."

In a February 11, 1861, editorial, the Madison, Wisconsin, Daily Patriot blamed "fire-eaters" on both sides—Southern Democrats and Northern Republicans. "Without reconciliation" between the two sides, then surely "war is inevitable".

Some commentators downplayed the risk of war, sometimes out of bluster, but mostly, out of an optimism that cooler heads would prevail once both sides recognized the dire consequences. "Will there be disunion?" the Indiana American of Indianapolis asked rhetorically in a November 21, 1860, editorial. They answered themselves:

"Of course not. The idea of a peaceable secession is too preposterous to be entertained for a moment by any sane mind, and no less absurd is the supposition that a feeble state like South Carolina, with neither men nor money nor credit, could ever seriously annoy the nation by an attempt at forcible disunion. It does not mend the matter to combine five or six states of like character. They all have not the means within themselves of keeping their negroes in subjection in such a contest, much less of seriously annoying the nation."

On January 1, 1861, the Philadelphia Public Ledger summarized the sentiment of the day in the introduction to an editorial:

"In the present temper of the public mind North and South, it seems to have been lost sight of that the only means by which this excitement can be allayed, the authority of the government restored, and the country saved from civil war, is by conciliation and concession. The public mind is fast drifting into the idea that a collision is inevitable between the two sections of the country, the North and the South; and as this idea seizes it more strongly, there is defiant language, bitter taunts, a disposition to shut out all compromise, and preparation for war."

Once Lincoln had become President-Elect, it wasn't just predictions and accusations of who was in the wrong, and what was to be done to stop war, that was dominating political discourse. The (soon-to-be) Confederate leadership was essentially promising to wage war if their attempt at secession were to be opposed by the federal government. The pre-election rhetoric spread beyond just the most radical Fire-Eaters, but began to gain support among "moderate" Southern Democrats, as well as "Conditional Unionists" who had voted for the Constitutional Union Party in the 1860 election. They had been skeptical of secession as a political remedy before the election, but now were getting caught up in the furor and joining the fight. The smaller group of pro-secession, pro-war Fire-Eaters found new allies in their movement toward getting Secession Ordinances passed.

So, in the speeches of future Confederates in the period between November 1860 and April 1861, it's common to see these Southerners promise a war if the U.S. federal government opposed their movement. As a few examples:

On January 7, 1861, Gov. Isham Harris of Tennessee gave a speech to a special session of that state's General Assembly who were considering holding a Secession Convention. Harris promised war if secession was opposed by the federal government:

".... hopeful as we may be of finding some remedy for our grievances consistent with the perpetuity of the present Confederacy [i.e., at that point, this was his word for the United States], the question, at last, is one which each member of the Confederacy must determine for itself, and any attempt on the part of the others to hold, by means of military force, an unwilling sovereignty as a member of a common Union, must inevitably lead to the worst form of internecine war..."

On January 10, 1861, Jefferson Davis gave a speech on the floor of the U.S. Senate, admonishing the U.S. government for not having evacuated Fort Sumter. While he made some statements hoping for peace, he promised war if the federal government would not agree to recognize secession:

"The time is near at hand when the places which have known us as colleagues laboring together, can know us in that relation no more forever. I have striven to avert the catastrophe which now impends over the country, unsuccessfully; and I regret it...If you desire at this last moment to avert civil war, so be it; it is better so. If you will but allow us to separate from you peaceably since we cannot live peaceably together, to leave with the rights we had before we were united, since we cannot enjoy them in the Union...

"If you will not have it thus; if in the pride of power, if in contempt of reason and reliance upon force, you say we shall not go, but shall remain as subjects to you, then, gentlemen of the North, a war is to be inaugurated the like of which men have not seen."

On March 2, 1861, Gov. Harry Rector of Arkansas opened that state's Secession Convention with a speech that promised war if secession was met by federal opposition:

"All confidence is lost and it is too late to repair it. The honor and sensibilities of southern men have been trampled upon and wounded until the two sections stand in undisguised antagonism. What house can stand divided against itself? Let us then separate in peace if possible; if it not, then let it be in war, for separation must come sooner or later, and our danger increases in magnitude."

Thus, when the federal government refused to vacate federal property at Fort Sumter, and the South Carolina militia attacked, and then when Abraham Lincoln consequently called for all the states' militias to aid in putting down the insurrection, the Confederates responded with, "See? He made us do it! Opposition to our militia-backed secession is an act of war, so now there's war, and the Republicans started it!"

cont'd...

I'd like to add to the question: what was the view of the situation outside of the US? Did the US civil war seem likely to for example French, Spanish or English diplomats and politicians?

Just to add a bit to the answer of /u/secessionisillegal, as they mention it's important to remember the history of the threat of secession long preceded November 1860. Andrew Jackson has fallen dramatically out of favor in the last several decades for some pretty good reasons, but part of why he's on the $20 bill and Democrats named their annual party dinners Jefferson-Jackson for well over a century was not just his (rather highly politically motivated and slightly overstated) work to finish eliminating property requirements for voting for white males, but also his vigorous response that kept the Union together.

An attempt to ambush Jackson was made in 1830 at a Pennsylvania Democratic party dinner (just Jefferson Day back then) that was stacked with those who supported the nebulous concept of "states rights" that descended from the Kentucky and Virginia resolutions. It was thought at the time that Jackson had supported that concept, but he got wind of it - as did a number of other Northern politicians who avoided the dinner like the plague - and gave one of the most important toasts in American history.

"Our Union! It Must and Shall* Be Preserved!"

(*Sources differ on if 'and Shall' was actually part of the toast, although the Northern rallying cry in the decades to come used it.)

The threat of secession by South Carolina a couple years later was very real, but Jackson's vigorous opposition and reputation of using military force was a significant reason that a Civil War didn't break out at a time when the general outlook would have been far less favorable for the North than 3 decades later.

But to get back to the 1860 time frame, something that doesn't get noted much about the nominal offers of compromise at the time made by Unionists in the South were that their demands weren't just that Fort Sumter and Fort Pickens not be reinforced (which actually leads to a funny story about Lincoln, Seward, and Wells stepping on each other's toes with the result that the same ship received separate sets of orders to relieve both forts) - but that the terms and conditions of peaceful coexistence included the demand that the Union would cease collecting tariffs at all Southern ports.

Not reinforcing or only providing provisions to Fort Sumter and the three other Southern forts was one thing, but taking away something like a third of the revenue of the Federal government (which on top of everything else was basically broke at that point) and providing it to finance a rebellion was another. Seward actually had hope for the first six weeks or so following inauguration - when it can be argued that he was more influential than Lincoln and was very open to drastic steps to keep the South from secession - that the various compromise measures, including the Corwin amendment supported by both him and Lincoln (and for that matter Buchanan) had a shot to keep the South in the Union. However, Seward basically gave up once he finally met with Virginia Unionists and realized the tariff demand among others was one that was going to be simply impossible to meet.

So yes, once Lincoln had been elected the fear of Civil War was very real - and very justified.