Tevez's advisor Kia Joorabchian is reported to be in talks with Paris St Germain over a potential move to the French club, while Inter Milan have had a bid rejected for the former Manchester United star and AC Milan recently pulled out of the race to sign him.
The 27-year-old has not played for City since September following his touchline row with Mancini during a Champions League game at Bayern Munich when he allegedly refused to play as a substitute.
Tevez is currently back home in Argentina awaiting a decision over his future and, with the transfer window shutting in less than two weeks, Mancini is keen to see the matter sorted out once and for all.
"I hope this will finish very quickly. I don't know the latest," Mancini said on Friday.
"I think he has talked with PSG and Inter, but not AC Milan. I hope for him everything can finish very quickly."
Mancini was asked if Tevez could return to the City team if a move fails to materialise before the end of the January transfer window.
The Italian made it clear that would not be allow to happen and said: "This is impossible. Carlos doesn't want to stay.
"It is important for him to start to play because he is a football player.
"It is now three months that he hasn't played and this is not good for him."
Meanwhile, City defender Nedum Onuoha is edging closer to a move to QPR.
The 25-year-old appeared as a substitute in the Premier League leaders' win at Wigan on Monday, but has otherwise struggled for opportunities since returning from a loan at Sunderland.
Mancini said: "They were talking yesterday (Thursday). Maybe today or tomorrow, it (a transfer) is possible.
"Nedum deserves to play. He is a good guy and if he stays here he will maybe play one or two games. That is not enough for a young player."