Looks like we may have already had both: President James Buchanan and Vice-President William Rufus King.
From the Chronicle of Higher Education:
Buchanan’s long-time living companion, William Rufus King, was referred to by critics as his “better half,” ‘’his wife,” and “Aunt Fancy.” Around Washington, the pair were known as the “Siamese twins,” slang at the time for gays and lesbians.
King was vice-president under Pierce, who served immediately before Buchanan. Incidentally, while Buchanan was the only bachelor president, King was the only bachelor vice president.) Updike’s conclusory analysis that the two of them would never dare to consummate their relationship is, however, not persuasive. Other authors have noted that Senator King was called “Miss Nancy” by Andrew Jackson, “Mrs. James Buchanan” by James K. Polk’s law partner, and “Buchanan’s better half” and “Aunt Fancy” by others. Senator King was noted for his “fastidious habits and conspicuous intimacy with bachelor Buchanan.” It’s not clear whether the cruel jibes were based on the facts of the Buchanan-King friendship (meaning they were gay) or if the two of them stayed fast friends precisely because of the bullying from other frontier politicians (meaning that they might not have been). It’s also possible that perhaps King was gay, and Buchanan was not, and that was what led to their intense friendship.