The track was released in April (14) and the rockers alerted their publishers to similarities between Smith's single and their song, which was performed by Tom Petty and The Heartbreakers.
According to Britain's The Sun newspaper, both parties settled the copyright dispute out of court in October (14) and a representative for Smith has now confirmed the news.
He says, "Recently the publishers for the song I Won't Back Down, written by Tom Petty and Jeff Lynne, contacted the publishers for Stay With Me, written by Sam Smith, James Napier and William Phillips, about similarities heard in the melodies of the choruses of the two compositions.
"Not previously familiar with the 1989 Petty/Lynne song, the writers of Stay With Me listened to I Won't Back Down and acknowledged the similarity... Although the likeness was a complete coincidence, all involved came to an immediate and amicable agreement."
The spokesman did not confirm if Petty and Lynne will receive royalties for the track.
Bill Freimuth from The Recording Company, which hosts the Grammy Awards, said the rockers will not be considered nominees for Stay With Me, which is up for three awards, because they "did not do any new writing" for the single.
He adds, "Rather, they would be given certificates to honour their participation in the work, just as any other writers of sampled or interpolated work (if the song wins)."