tl;dr
Nope. The straights can keep him.
Deets
There is no evidence that Burgess was homosexual or bisexual. In the first volume of his memoirs Little Wilson and Big God, he writes of Earthly Powers:
I wrote a novel some years ago which presents a whole lifetime of homosexuality and, in American bookshops, found its way to the shelf specializing in "gay" literature. For all that, I have never had homosexual proclivities, and I do not well understand what causes the inversion, which goes against biology. (p. 387)
He also mentions that during his time in Malaya, he had a homosexual cook named Yusof who came on to him with no success:
He was homosexual but far from effeminate. ... He could shift a piano single-handed. He dyed his hair with henna and muscularly minced. The advances he made to me were politely repelled. (p. 379)
The second volume of his memoirs, You've Had Your Time, repeats Burgess's claims that Earthly Powers was not based on his own experience, and that homosexuality held no attraction for him:
I proposed some day to write a novel from the viewpoint of a homosexual and achieved this in 1980. Apparently the impersonation worked, for the book found its way to the shelves of Gay Lit in American bookstores. Lynne could believe what she wished, but I have always been afflicted with a powerful but banal heterosexual drive, unmodified by the sight of Greek or African boys lying naked in the sun. (p. 116)
The "Lynne" mentioned is Burgess's second wife, who apparently kept setting Burgess up with her homosexual friends and insinuating that he must be homosexual. At one point she explicitly asks him whether he is a closet homosexual, which he denies. (p. 70)
Of course there is no guarantee that Burgess, a notorious fabulist, is telling the truth about any of this. And someone's saying "I'm not gay" is no evidence of their heterosexuality. But two biographies of Burgess (by Roger Lewis and Andrew Biswell) also make no mention of any relationships Burgess might have had with men. Lewis in particular is scathing about every aspect of Burgess's personality, and he would have had relished the opportunity to reveal Burgess's love affairs with men. Instead, we get this footnote:
Burgess was fascinated by homosexuality (Toomey, Marlowe, the state-enforced faggotry of The Wanting Seed, the way homosexuality is a metaphor for the similitude and interpenetration of East and West in Honey for the Bears—a notion foreshadowed in Time for a Tiger, where the East is called "a horrible sweating travesty of Europe"), and yet he denies this vociferously to Duncan Fallowell: "I wish I could approve of homosexuality, but I'm enough of a Catholic to regard homosexuality as an aberration, as the spending of seed in barren places ... I don't know why the hell it exists, you see. It must be a genuine aberration, it's not natural. I've just been reading Aldous Huxley's essay about parrots, which imitate human speech, though they don't have the apparatus and there's no earthly biological reason for it. ... This is like homosexuality. What is nature up to here? Only God would be interested in playing such games with nature. In making parrots speak ... or homosexuality." (p. 229)
(Toomey is of course the narrator of Earthly Powers; notorious homosexual Christopher Marlowe is the protagonist of A Dead Man in Deptford.)
The depiction of homosexuality as unnatural, and its association with biological as well as metaphorical sterility, is representative of Burgess's Catholic attitudes. His fiction also depicts homosexuality in negative ways. For example, The Wanting Seed (1962) is a dystopia where heterosexuality is proscribed and homosexuality used as a means of population control. Earthly Powers, with its unlikeable, pompous, and petty narrator, is no exception to Powell's snide moralizing. The casual homophobia evinced throughout Burgess's œuvre (fictional as well as autobiographical, insofar as the two can be considered distinct categories in his case) is perfectly in keeping with times. (The homophobia of Lewis's footnote is another matter, given that his biography was published in 2002.)
The fan blog In Search of Anthony Burgess points out another piece of negative evidence, to wit, nobody has ever claimed to have had a homosexual encounter with the author:
Burgess’s own self-assessment, that he had ‘always been afflicted with a powerful but banal heterosexual drive, unmodified by the sight of Greek or African boys lying naked in the sun’, can be accepted as accurate, especially in view of the fact — as a commenter points out below — that in the 20 years since Burgess’s death not a soul has come forward to say he had any kind of homosexual encounter with him, even in childhood.
Those who identify Burgess as homosexual or bisexual on the basis of Earthly Powers (or his other novels where homosexuality is a major theme, such as A Dead Man in Deptford, The Wanting Seed, or Honey for the Bear) assume that only homosexuals would write about homosexuality. This is rather an archaic mindset, the same one that up until around twenty years ago made putatively heterosexual Hollywood actors hesitate to accept any role where the character was homosexual. But even then, there were straight actors who portrayed gay characters, such as Michael Caine and Christopher Reeve in Deathtrap (1982, right around the same time as Earthly Powers). Burgess, writing about homosexuality and even using a first-person homosexual narrator in 1980, might have been pioneering in the same way as Caine and Reeve. But that doesn't make him queer as a clockwork orange. Nor does it even mean that his representations of homosexuality were gay-friendly, any more than Reeve's and Caine's portrayal of a couple who conspire to murder Caine's wife is a triumph of representation for sexual minorities. Like the actors, Burgess gets to pat himself on the back (and even gets plaudits from others) for his daring even while perpetuating unoriginal, lazy, and pernicious stereotypes about homosexuality.
References
British Library. Accusations against Christopher Marlowe by Richard Baines and others. From "Discovering Literature: Shakespeare & Renaissance collection items". Retrieved from www.bl.uk on 13 May 2023.
Burgess the Gay-Lit God. Blog post on In Search of Anthony Burgess, Retrieved from burgessodyssey.wordpress.com on May 13, 2022.
Burgess, Anthony. Little Wilson and Big God: Being the First Part of the Confessions of Anthony Burgess. 1987. London: Penguin, 1988. Retrieved from archive.org 13 May 2023.
Burgess, Anthony. You've Had Your Time: The Second Part of the Confessions. 1990. New York: Grove, 1991. Retrieved from archive.org 13 May 2023.
Lewis, Roger. Anthony Burgess: A Biography. 2002. New York: Thomas Dunne Books, 2004. Retrieved from archive.org 13 May 2023.
McCrum, Robert. Unearthly Powers. Review of The Real Life of Anthony Burgess by Andrew Biswell. The Guardian, 6 November 2005. Retrieved from theguardian.com 13 May 2023.
Morrison, Blake. Kingdom of the Wicked. Review of Anthony Burgess by Roger Lewis. The Guardian, 8 November 2002. Retrieved from theguardian.com 13 May 2023.