The short answer is: possibly, but there’s no definitive evidence.

The longer answer is: James Buchanan was our only lifelong bachelor president. He was engaged once to Anne Coleman as a young man, but she broke off the engagement and died shortly thereafter. Rumors about dishonest motives for marrying haunted him and may have contributed to her breaking it off. The rumor mill intensified after her father refused to allow Buchanan to attend her funeral. He never courted another woman, so far as we know. There was a rumor of an affair with Sara Childress Polk—the widow of President James K. Polk—but these are unsubstantiated.

The homosexuality rumor arose later. Buchanan developed a deep, intimate relationship with William Rufus King. Their friendship is largely central to the debate. The letters between the two men are intimate without being sexual or explicitly romantic. They lived together—a not uncommon arrangement at that time—and spent much of their lives together. President Andrew Jackson referred to King as “Miss Nancy” and he was also called Buchanan’s “wife,” though political mudslinging could be behind this as well (19th century politicians were notoriously catty gossips).

Sadly, King’s family destroyed the bulk of their letters upon his death, so there’s little firm evidence to support the conclusion that he was gay. Is it possible? Yes. If so, he was closeted and took pains to remain so. In all likelihood, he was celibate. He may have been asexual. We can speculate endlessly. Part of the issue is homosexuality was largely criminalized then, or at the very minimum shunned. The rumors clearly didn’t kill Buchanan’s career (King died four years before he became President), but they did follow him all his life. As such, there may never emerge any strong evidence to prove the matter.

Here’s an article by sociologist and historian James Loewen on the subject:

https://www.salon.com/test/2012/05/14/our_real_first_gay_president/

Also, for reference: Baker, Jean. 2004. James Buchanan. Times Books.