The Welshman has been frozen out at the Etihad Stadium and spent last season at hometown club Cardiff but failed to lead them to promotion.
He doesn’t figure in Mancini’s plans for the coming season with the Italian telling him he can leave.
He told BBC Sport: “If Celtic want Craig… he can go. Craig is a good striker and he can play at Celtic, because they’re an important team.”
Neil Lennon is interested in bringing Bellamy back to Scotland after he enjoyed a superb spell with the club at the back end of the 2004/05 season winning the Scottish Cup in the process.
The Celtic boss is hopeful that he can bring Bellamy in before the transfer window closes and admits the two clubs have been in contact.
“There have been conversations between both clubs.” He told BBC Sport
“I didn’t speak with City, but the two clubs have been working on that in the background,”
“Hopefully there will be more to report at the end of the week.”
The 32-year-old has one year remaining on his current deal at Eastlands and would be prepared to see that out North of the border provided they match his wage demands.