The song, Home To Us, was released on Friday and is part of Sir Paul's forthcoming solo album.
The track also features Chrissie Hynde and Sharleen Spiteri on backing vocals.
The new album, titled The Boys Of Dungeon Lane, recounts a collection of memories from the singer's formative years in Liverpool, including his early adventures with his late bandmates George Harrison and John Lennon, prior to their Beatles fame.
Sir Paul named the introspective 14-track record after Dungeon Lane, a place close to his childhood home in Forthlin Road in the Liverpool suburb of Speke.
The album, his first in five years, is due to be released on May 29, with Sir Paul previously describing the inspiration behind it as "a lot of memories of Liverpool".
"We didn't have much at all but it didn't matter because all the people were great and you didn't notice you didn't have much," he added.
The record will also feature a number of new love songs and reflections from the artist on his life before he found global fame.
He released lead single, Days We Left Behind, in March.
Sir Paul's new studio album was produced by Andrew Watt and began nearly five years ago with the opening track, As You Lie There.
Former Beatles drummer Sir Ringo released his latest solo album last month.
The 10-track record, Long Long Road, features collaborations with Billy Strings, Sheryl Crow and St Vincent, and was co-written with American producer T Bone Burnett.
It followed on from Sir Ringo's 2025 album Look Up, which marked his first number one album on the UK's official country chart.
The Beatles were formed in 1960, comprising Sir Paul, Lennon, Harrison and Sir Ringo.
The band became the best-selling musical act of all time and one of the most influential groups in the world, having achieved 16 UK number one albums and 18 UK number one singles including Help!, Hey Jude and She Loves You.
Following their split in 1970, Sir Paul released his debut solo album, McCartney.
He also formed the band Wings with his late wife Linda McCartney along with former Moody Blues guitarist Denny Laine.
Prior to splitting in 1981, the band recorded seven studio albums including Band On The Run and achieved 12 UK top 10 singles including Jet, Silly Love Songs and Live And Let Die, the theme to the 1973 James Bond film.
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