Well, it was definitely believable, at it was not entirely a rumor!

In September, 1861, Henry Shelton Sanford, head of the US mission in Belgium, traveled to the island of Caprera to offer Giuseppe Garibaldi a commission as a general in the US Army. He was fresh off of his victory at the head of the so-called "Thousand" which took on the Kingdom of Two Sicilies., and one of the most famous military commanders in the world. Far from there being a reaction to the invitation, the invitation itself was a reaction to his popularity, as there had been a noticeable cry for his coming to the Union cause all through the summer, with newspapers variously reporting on whether or not rumors of his impending arrival were true. Lincoln's decision to attempt to make use of his services were thus very much a ploy to court the current trends, and win a public relations victory. As it would turn out, the rumor mill had been started by Garibaldi himself, who had instructed his secretary, Candido Vecchi, to drop some teasing hints to the press, who was more than happy to run with it.

The rumors began to percolate into more concrete possibilities in June, when James W. Quiggle, an official at the US Embassy in Belgium, wrote to the Italian leader, began a correspondence concerning Garibaldi's possible service, and even offering to join him as an aide. After a exchange of letters, the he duly sent it on to Washington as evidence of Garibaldi's willingness to serve, arriving mere days after the disastrous battle of Bull Run. Good timing for Garibaldi though, as it meant that Seward and Lincoln looked quite favorably on the missives, and instructions were duly sent back to Belgium, although the newly appointed Sanford would be handling it from here on out. Seward's instructions can be summed up as the following:

I wish you to proceed at once and enter into communication with the distinguished Soldier of Freedom. Say to him that this government believes his services in the present contest for the unity and liberty of the American People would be exceedingly useful, and that, therefore they are earnestly desired and invited. Tell him that this government believes he will if possible accept this call, because it is too certain that the fall of the American Union . . . would be a disastrous blow to the cause of Human Freedom equally here, in Europe, and throughout the world.

Seward's meeting did not, however, result in Garibaldi accepting the proffered position. In the days leading up to the meeting, Sanford did his best to try and gauge the likelihood Garibaldi would accept. While the impression given had been one of gusto, discussions with his counterpart in Italy George Marsh, gave credence to the possibility that Garibaldi had set up the entire ordeal to give hi leverage with King Emmanuel II, and get the go-ahead to march on Rome. Just prior to their meeting, Garibaldi did, in fact, attempt to force the King's hand, but was rejected, so Sanford arrived at Caprera fairly confident that the great Italian would now accept the offer. It was not to be. Perhaps Garibaldi never intended to accept, and after the King had called his bluff now needed excuse to pass on the Union offer, or perhaps his offense was simply real, but in either case, when Garibaldi learned he was being offered a command, instead of overall command (perhaps where your confusion in the title is?), he balked. He offered further reticence when he was informed that the war (at that time) was not to end Slavery, which Lincoln disavowed the power to do - although this had been apparently made clear in Quiggle's letters.

Seward departed, and the public explanation for the failure of Garibaldi's acceptance was given as the disagreement on rank, with his disappointment that the immediate end of slavery was off the table consigned only to private memos to Seward. And although in later correspondence, Garibaldi repeating this to a friend "You may be sure that had I accepted to draw my sword for the cause of the United States, it would have been for the abolition of Slavery, full, unconditional" he also kept his public statements cordial, preferring to keep the possibility of joining later open, and would continue to issue public statements in support of Union.

A year later, another attempt would be made, this time by Heinrich Theodore Canisius, a consul in Vienna. He wrote to him, without any authorization from Washington:

As you have failed for the present to accomplish the great and patriotic work you lately undertook in the interest of your beloved father land, I take the liberty to address myself to you, to ascertain whether it would not be against your present plans to lend us a helping hand in our present struggle to preserve the liberty and unity of our great Republic. the battle we fight is one which not only interest ourselves, but also the whole civilized world.

Garibaldi had recently been wounded, and taken prisoner following the Battle of Aspromonte, so in no condition to fight just then, but replied positively that hoped to help the Union when able. Canisius informed Seward at this point, but the papers as well, leading to another round of praiseful articles extolling the possibility of Garibaldi coming to America. Seward was less than pleased though, as Canisius had no authorization, and Garibaldi at that point was considered a rebel by Italy itself due to his having gone rogue. It was a propaganda victory for Garibaldi though, winning him praise from all corners, and opprobrium towards the Pope and the King. He returned the favor later that month in his public letter "To the English Nation", written as a reaction to the public protests against his imprisonment staged in the UK, and in which he called on the British people to "Help [America], and then make her sit by your side in the great assembly of nations, the final work of the human reason." The letter did no small bit of good for the American cause, no doubt more than Garibaldi - "an aging Italian general who did not speak English, was severely crippled by recurring bouts of arthritis (he hobbled on crutches to greet Sanford), and had a proven record for insubordination" - could have achieved in service, giving strong and vigorous support for the Union that was echoed by the British people in return. Charles Francis Adams wrote to Seward from London that in the wake of the letter “Less and less appears to be thought of mediation or intervention [and] all efforts to stir up popular discontent [against the Union] meet with little response. [...] I am inclined to believe that perhaps a majority of the poorer classes rather sympathize with us in our struggle and it is only the aristocracy and, the commercial body that are adverse.” Back in Italy, although they lacked the same public demonstrations as in England, fears of an uprising led to the King freeing the imprisoned general.

That would be the last serious effort to bring Garibaldi into the Union blue. George Marsh would send one last message of encouragement in October, 1862, but there were no more after that. The following spring, after the Emancipation had finally made ending slavery, Garibaldi would send off a public letter, praising Lincoln that “An entire race of men, bowed by selfish egotism under the yoke of Slavery, is, at the price of the noblest blood of America, restored by you to the dignity of man, to civilization and to love,” but he wouldn't seek any active role. Nevertheless though, he had played an important role, if not on the field, then in the press. Association of his name with the Union cause had certainly never done it ill, and later, his praise for the Union served it in very good stead. Doyle credits him with going to far as to playing a key role in preventing British and French intervention in that late 1862 period, which might be a stretch, but he without a doubt helped stir up popular support for the North abroad following his wounding and public letter to the English people.

Cited:

"The Cause of All Nations: An International History of the American Civil War" by Don H. Doyle

Not only was it believable that Garibaldi would be offered a commission, but in fact the offer was made - and declined.

In early 1861 a man by the name of Vecchi, a close friend of Garibaldi, was tasked with writing a thank you letter to Henry Tuckerman who had published an article praising the Italian general. Unbeknownst to Garibaldi, Vecchi included a suggestion that the General would be of great benefit to the Union cause if his services were requested.

News came out that Garibaldi had offered his services - which he hadn't. The U.S. Consul in Antwerp, James Quiggle, sent a letter requesting clarification. Garibaldi responded that he "would go to America" if Italy was secure, and also "whether this agitation is the emancipation of the negroes or not." Quiggle's response was to the negative, and he forwarded their correspondence to the State Department in Washington.

This news arrived shortly after the Union defeat at Bull Run, and in the ensuing upset Secretary Seward assigned Henry Sanford, newly appointed Ambassador in Brussels, to inform Garibaldi that his services would be "exceedingly useful, and that, therefore, they are earnestly desired and invited," adding that "he will receive a Major-General's commission in the army of the United States."

After a series of correspondence and travel, Sanford finally met with Garibaldi in order to make the offer. Ultimately the offer was met with refusal. Garibaldi still held aspirations for marching on Rome, and that the only way he could assist the cause of the Union "was as Commander-in-Chief of its forces," and if given the authority of "declaring the abolition of slavery". At this point the U.S. could offer neither of the latter, which left Garibaldi to attempt achieving the former.

In 1862 Garibaldi committed himself to a campaign to capture Rome, in violation of his government's wishes. A short time later he was arrested, and as soon as word reach the U.S. Consul in Vienna, an unauthorized offer was made to Garibaldi - who was forced to refuse given his current situation.

As an aside, August 6th of the following year, Garibaldi wrote to Lincoln regarding the Emancipation Proclamation:

"In the midst of your titanic struggle, permit me, as another among the free children of Columbus, to send you a word of greeting and admiration for the great work you have begun. Posterity will call you the great emancipator, a more enviable title than any crown could be, and greater than any merely mundane treasure. You are a true heir of the teaching given us by Christ and by John Brown. If an entire race of human beings, subjugated into slavery by human egoism, has been restored to human dignity, to civilization and human love, this is by your doing and at the price of the most noble lives in America.

It is America, the same country which taught liberty to our forefathers, which now opens another solemn epoch of human progress. And while your tremendous courage astonishes the world, we are sadly reminded how this old Europe, which also can boast a great cause of liberty to fight for, has not found the mind or heart to equal you."

Transcripts of most of these correspondents are available in H. Nelson Gay's 1907 article "Lincoln's Offer of a Command to Garibaldi."