‘Angel In Disguise’ was written by the pair for Starr’s 1992 solo album ‘Time Takes Time’, but the demo did not make the LP’s final cut.
It is now being sold by former Radio Luxembourg DJ Tony Prince and is expected to fetch up to £20,000 at auction.
A quarter of the profits will go to the NHS Charities Together Covid-19.
The demo tape features two versions of the track – a demo with McCartney on vocals and a fuller take featuring Starr’s vocals, alongside additional instruments and backing vocals.
A demo of another Ringo track, called ‘Everyone Wins’, also features on the cassette, which is being sold by Omega Auctions on May 19, alongside other memorabilia.
The sheet music for the track, which credits McCartney as the only writer, includes lyrics such as: “My name is Ritchie / Let me look into your eyes / Don’t be afraid I’m just an angel in disguise”.
“It’s fantastic to be able to bring to the market a previously unheard and unreleased track,” Omega auctioneer Paul Fairweather told the BBC.
“We think the track certainly had enough about it to be included in the album and I am sure if Paul was to release this himself it would certainly prove a hit with fans.”
Other items being sold at the online sale by Prince include a selection of letters and documents he was sent after the death of Elvis Presley in 1977, having broken the news live on air.
Autographs by Paul and his late wife Linda are also up for grabs.
Meanwhile, McCartney featured in Lady Gaga’s Together At Home concert last month, where he delivered a rendition of The Beatles’ ‘Lady Madonna’.
“I’m very honoured to be part of this programme tonight that celebrates the true heroes: our health care workers all around the world,” McCartney said before the performance.
“As this COVID-19 pandemic is a global crisis, we all have to come together to fight it on a global scale,” he added. “Let’s tell our leaders we need them to strengthen the health care systems all around the world so that a crisis like this never happens again.”