Depends on how you define "significant". The problem with the question is there is no modern-day scientific polling to draw from. The best that can be said is that, yes, briefly, before the Battle of Fort Sumter, there was some rhetoric in the North about letting the Confederates go.
Probably the most famous such statement was the one given by Horace Greeley, editor of the New York Tribune, the most widely read newspaper of the time. In an editorial published in the paper on November 9, 1860, just days after Lincoln's election and threats of secession had already been made (on the same day, the South Carolina state legislature passed their "Resolution to Call the Election of Abraham Lincoln as U.S. President a Hostile Act and to Communicate to Other Southern States South Carolina's Desire to Secede from the Union"), Greeley wrote:
"If the Cotton States shall become satisfied that they can do better out of the Union than in it, we insist on letting them go in peace. The right to secede may be a revolutionary one, but it exists nevertheless ... And whenever a considerable section of our Union shall deliberately resolve to go out, we shall resist all coercive measures designed to keep it in. We hope never to live in a republic whereof one section is pinned to the residue by bayonets."
But as both James Buchanan pointed out in his fourth and Final Message To Congress, and Abraham Lincoln did in his first Inaugural Address, "just letting the Confederacy break way and leave it at that" was much more complicated than it sounded. Not to delve too much into modern day politics, but it's the same issue the UK has found with Brexit. It's one thing to say "we're leaving". That's the easy part. The hard part is negotiating that exit.
Even if a requisite number of Congressmen and state legislatures granted permission, per the Constitution, to allow the Confederate states to leave, the issues to be negotiated remained the same whether the Confederate states stayed within the Union or they exited it, just as Buchanan and Lincoln mentioned. Most specifically:
The issue of "fugitive slaves" had to be negotiated, either as domestic states under the Constitution, or as neighboring foreign countries.
The issue of slavery in the Western non-state territory of the United States had to be negotiated as domestic states or as neighboring foreign countries.
Further, secession brought up issues that wouldn't be pertinent if the states were still united:
How much of the national debt did the seceding states need to take on? They certainly couldn't stick the remaining states with the whole bill, since plenty of that debt had been taken on to benefit the South (some of it actually being land and other properties the federal government had bought in the South, so the South would need to return the money to pay for it).
Who got to keep Washington, D.C.? It was on the Maryland side of the river, but the Confederates definitely wanted to keep it, and maintain it as a slave state, er, district. How much was it worth and how much would the Confederates be willing to pay for it?
How would control of the borders be handled? Specifically, who controlled what portions of the Ohio and Mississippi Rivers, and how would slavery be handled in them? Union states would be sharing borders with slave states along those rivers. Was transportation of enslaved people going to be allowed? Would this be a stipulation for the North to gain access to New Orleans? But then again, the North had another path to the ocean, via the Erie Canal and the Hudson River. How much did they really need to negotiate?
To put it bluntly, secession put the South in a worse negotiating position on all these issues. Without Southern votes in Congress, the North had no reason to have to negotiate on "fugitive slave" issues except that the South as a foreign country could threaten to attack with violence. Without Southern votes in Congress, the North had no reason to hand over Western territory or legalize slavery there except that the South as a foreign country could threaten to attack with violence. The North had no reason to hand over Washington, D.C., except that the South as a foreign country could threaten to attack with violence. The North had no real reason to pay back any debts owed to Southern states, and could direct other foreign countries to collect debts owed by the South directly from the South. The only recourse the South had was to threaten to attack the North with violence. The North could bypass the need for the Mississippi River, by relying on the Erie Canal route to the ocean instead. It was longer for some of the states, but the only recourse the South had for maintaining the legality of slavery on shared portions of the river was to threaten to attack the North with violence.
So, while there were calls before Fort Sumter that maybe the two parts of the United States should disunited peacefully, all these political issues had to be resolved one way or the other, either as domestic members of the same rule of law, or as foreign countries through treaties.
Ultimately, the South realized the only way to win the political negotiations that they wanted was through violence. This is why there was a war instead of negotiations. In fact, negotiations on many of these issues (Western expansion of slavery, the Fugitive Slave laws, and slavery in Washington, D.C.) were central to the peace efforts between Election Day of 1860 and the Battle of Fort Sumter of 1861.
Once the South opened fire on Fort Sumter, however, virtually all the calls in the North for a peaceful separation ended. Horace Greeley ended up being one of the most supportive voices of President Lincoln early in the war. "On To Richmond!" was the call of the newspaper in the months after Fort Sumter.
But how serious was this sort of "peaceful separation" outcome? Not very, according to David M. Potter in his 1941 article "Horace Greeley and Peaceable Secession". The negotiated outcome as a single united country was always more likely, and always more clearly understood, through domestic compromise than through treaty between foreign nations:
"Greeley's vociferation is a curious one. Insofar as the cause of voluntary separation depended upon him, it was a phantom from the beginning. If his attitude was typical (as historians assume), the go-in-peace program was devoid of appreciable support and never achieved a place in the realm of practicable solutions. It was, in this sense, altogether illusory and undeserving of the notice which history has given it. But, like other illusions and red herrings, it bore an enormous importance. For it constantly obscured the clarity of the true alternatives—compromise and war."
There did continue to be "Peace Democrats" who believed that some sort of peaceful resolution to the war was best. Perhaps chief among them were former President Franklin Pierce, U.S. House member Clement Vallandigham from Ohio, and U.S. Senator Jesse D. Bright from Indiana. However, their position was mostly from the standpoint of peaceful reunion, by the North being more willing to offer concessions to the South, with disunion being a last resort. If it's any indication, when Vallandigham insisted on making Confederate-sympathizing statements, he was arrested and "deported" to the Confederacy. He immediately surrendered to Confederate troops as a U.S. citizen. He was briefly imprisoned, but then allowed to "escape" to Canada, from where he ran for Governor of Ohio as a Democrat (and got trounced). He may have been willing to recognize the Confederacy as some foreign power, but he was not really interested in actually being a Confederate citizen. Instead, he wanted to see the two sides come to an agreement, hopefully one that kept the country together ("unconditional unionist"), not one that kept it apart.
One other point to make, which complicated things. If the U.S. acquiesced to the demands of the South, they also had the West to deal with. In California and Oregon, there were vocal government officials who were calling for their own Western country, separate from either the North or South, if disunion should be accomplished. Once again, it would be a negotiation with three different countries on the future of slavery, as well as border issues and other issues, which had to be resolved one way or the other. Either through domestic negotiations in Congress, or as foreign countries through treaties.
FURTHER READING:
Bonner, Thomas M. "Horace Greeley and the Secession Movement, 1860-1861", The Mississippi Valley Historical Review, 1951.
McClintock, Russell. Lincoln and the Decision for War: The Northern Response to Secession, 2008.