Sinclair moved to the Etihad Stadium from Swansea City in August but he has only featured 14 times for City since his arrival.
His former club have registered an interest in re-signing the 24-year-old, and Mancini has revealed that he could be prepared to allow the former Chelsea trainee to leave the club.
The 48-year-old told reporters: "This situation with Sinclair is difficult because I didn't give him a lot of chances to play. He is very unlucky. It is my fault, not his fault. A young player like Scott who is a good player deserves to play always.
"I think it is better if he can have another solution. If he stays here, maybe, everything can change but I can understand his frustration in this moment because he didn't play a lot of games."
Mancini was speaking ahead of City's trip to the Liberty Stadium to face Sinclair's former side.