The band have revealed they were asked to write a signature tune for the movie, but it "didn't work out" and the song was not used.
Sam Smith's Writing's On The Wall ended up as the movie's theme song instead.
Radiohead have now made their brooding, orchestral song, titled Spectre, available as a free download on Soundcloud.
Singer Thom Yorke wrote on Twitter: "Last year we were asked to write a theme tune for the Bond movie Spectre.
"Yes we were. It didn't work out, but became something of our own, which we love very much.
"As the year closes we thought you might like to hear it. Merry Christmas. May the force be with you."
Many fans expressed their approval on Twitter. Music critic Pete Paphides was among them, writing: "It sounds exquisite on first listen, sensational string arrangement."
Before Spectre came out in October, there had been speculation that Radiohead had been chosen for the coveted task of recording the theme tune.
In July, one bookmaker suspended betting on the identity of the act that had recorded the theme after one punter tried to bet Â£15,000 on Radiohead.
Bond actor Daniel Craig is known to be a fan of the band, having quizzed Yorke for Interview magazine in 2013.
Sam Smith's theme received mixed reviews when it was released in September, with BBC entertainment correspondent Colin Paterson saying it was "good enough, but not a classic".
But former Bond actor Sir Roger Moore called Writing's On the Wall "very haunting and wonderfully orchestrated".
The song made history by becoming the first Bond theme to reach number one in the UK, achieving combined chart sales and streams of 70,000 copies.
Smith was the first solo male British act to perform a Bond film's theme song since 1964, when Tom Jones was heard over the opening credits of Thunderball.
Radiohead have released the song they recorded to be the theme for the latest James Bond movie Spectre.