Tormé, who toured with Osbourne in the early eighties, died on Sunday, a month after an update posted on his Facebook page announced that he was being treated for double pneumonia.
Osbourne shared a message honouring Tormé’s memory after the news of his death broke, tweeting: “What a sad day. We’ve lost another great musician.
“Bernie was a gentle soul with a heart of gold. He will be dearly missed.
“I send my sincere condolences to his family, friends and fans. Rest in peace Bernie.”
Tormé joined Osbourne on his Diary of a Madman Tour in 1982, following the death of guitarist Randy Rhoads in a plane crash that same year.
He had previously performed with Gillan, the band formed by Deep Purple lead singer Ian Gillan.
Osbourne and his wife Sharon Osbourne had sent Tormé a message of support when it was first reported that the musician was suffering from pneumonia.
“Dearest @Bernie_Torme, sending you positive vibes and love, and all our prayers,” they wrote in February. ”Love Ozzy and Sharon.”
Born in Dublin in 1952, Tormé rose to fame in the seventies, moving to London in 1974.
His touring stint with Osbourne saw him perform at New York City's Madison Square Garden. Tormé then released his first solo album in 1952.
He kept releasing records throughout the years, either solo or with bands. His final release was the album Shadowland in 2018.