The oil painting, depicting the singer as a melancholy clown, is believed to have been painted in 1957 during his divorce from Gardner, his third wife, whom he once described as "the only woman I've ever been in love with my whole life".
It is one of more than 30 artworks by musical legends including Sir Paul McCartney, John Lennon, Johnny Cash and Kurt Cobain that will feature in a new exhibition at the Scream gallery in west London.
The Sinatra self-portrait, which will go on sale later this month for £25,000, was inspired by the 1957 film The Joker is Wild, based on the life of Sinatra's friend, Joe E. Louis.
Sinatra was said to have taken the lead role only because it was being filmed in Spain where Gardner lived following their separation, and where he hoped to dissuade her from a divorce.
The couple were married from 1951 to 1957, but their turbulent love affair was riddled with jealousy, alleged affairs and heavy drinking.
The leader of the Rat Pack gave the self-portrait to his friend Sonny King as a gift in 1964 and it has been in private collections ever since.
Sinatra, who died in 1998 aged 82, enjoyed painting as a hobby for more than 40 years.
He had a studio in his home, published a book of his abstract works, and sometimes spoke of his artistic endeavours in a self-depricating way during concerts.
In 1958, the year after Sinatra and Gardner divorced, another painting of Sinatra as a tragic clown, by the celebrity portraitist Nicholas Volpe, was used on the cover of Sinatra's album Only the Lonely.
The exhibition - curated by Jamie and Tyrone Wood, the sons of Ronnie Wood, the Rolling Stones band member - will also feature several later abstract works by Sinatra which have never been displayed before, and an ink sketch of a steam train that the singer drew on a tablecloth at London's Mirabelle restaurant in 1986.
Brandon Coburn, an art consultant and co-curator of Original Artwork by Pop Culture Artists, which opens on September 24, said: "These bold and expressive paintings are a true example of Frank Sinatra's deeper interests and emotions as well as his ability to paint and entertain in an alternative manner to singing.
"The works are reflective of why Sinatra was driven to paint. Unlike performing, which was publicly exhilarating, painting became his favourite way to relax and restore his energy."
The exhibition also includes works by Lennon and McCartney, completed before and during their time in The Beatles.
They include two drawings by Lennon and sketches by a teenage McCartney, drawn while he was a student at the Liverpool Institute in the 1950s.
One of McCartney's sketches depicts Elvis Presley floating above a bucolic scene, while another shows two young women and a man in sunglasses with the inscription "Nurse Young".
Another pencil sketch shows a young boy and a woman, with the title "And it was me all the time, Aunty". The works by McCartney are believed to be on sale for between £35,000 and £50,000.
Two previously unseen paintings by Cobain, the troubled lead singer of the rock band Nirvana, will also feature in the show.
Completed during the early 1980s when he was a student at Aberdeen High School in Seattle, the airbrush paintings show fluorescent abstract landscapes, with one featuring an illustration of a guitarist.
During the last two years of his life, the singer is said to have spoken frequently of abandoning music for a career in visual arts, an ambition that he never realised. He committed suicide in 1994, aged 27.
Jamie Wood, 35, who runs the Scream gallery with his brother Tyrone, 26, said that he hoped the exhibition would establish the likes of Sinatra, McCartney and Cobain as "great visual artists in their own rights".
He said: "There is definitely a snobbery in the art world about work by celebrities who aren't deemed true 'artists'.
"When I started selling Dad's work 10 years ago, the so-called connoisseurs weren't interested because he wasn't an established painter, but now his works sell for more than half a million pounds because people recognise that he is an accomplished artist.
"These pictures, from popular cultural icons past and present, actually show so much more emotion than we see in much contemporary art these days, because they are by multifaceted artists whose creativity extends beyond just one dimension.
"In 100 years time, what will the collectors really want to own as great works and investments - an ashtray by Damien Hirst or a Sinatra self-portrait?"
Other works that will go on display include portraits of Jimi Hendrix and Bob Dylan by Grace Slick, the former Jefferson Airplane singer, sketches and paintings by Ronnie Wood, illustrated lyric sheets by Joni Mitchell, a series of surreal drawings by Johnny Cash, and portraits by Volpe of Sinatra, Ronald Reagan and Sammy Davis Jr.
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