The 80-year-old said he was "fuming and furious" about the error, which saw songs by the group Abhorrent Decimation printed on his CD.
The singer made an album of covers following the response to his audition on BBC One's The Voice.
He joked he should join the band "on the road", while their singer said they should "work together".
Clifton's track listing was mixed up with that of Abhorrent Decimation's album, Miasmic Mutation.
The comedian, who is famous for riding a yellow ostrich, said: "I'm fuming and furious... I got [the CDs] out the box and I looked on the back and here was the track listing of a death metal group.
"We could go on the road, me and the lads. We could do To Dream the Impossible Miasmic Mutation."
The band's singer Ashley Scott said he was alerted to the mistake after being shown an article about it on page three of The Sun newspaper.
"I thought he was going to show me some buxom blonde and take the mick out of me for being a bit overweight and bang, there's the band.
"I definitely want to talk to Bernie about working together on something... it would be amazing."
Mr Scott said he thinks the error occurred due to similar catalogue numbers.
Clifton, from Derbyshire, apologised to the people who pre-ordered his album and said copies would be with them shortly.
A list of tracks by a death metal band was accidentally printed on the back of a new album by comedian Bernie Clifton.