Tevez said in a statement released by City on Tuesday night that he was "sincerely and unreservedly" sorry for his actions in taking an unauthorised three-month leave of absence from the club.
The Argentinian also withdrew his appeal against a fine of six weeks' wages.
Tevez has not played for City since he was accused of refusing to warm up during the Champions League defeat at Bayern Munich on September 27.
Mancini, at the time, said Tevez was "finished" at the club as a result and the dispute escalated when the player then flew to Argentina without permission on November 7.
However, after Wednesday night's 4-0 defeat of Porto booked City's place in the last 16 of the Europa League , Mancini said Tevez would be available once he deems him match fit.
The Italian told ESPN: "One week after Munich I called him to my house and said, 'Carlos, apologise to everyone and it's finished.'
"I have no problem. I think he needs two to three weeks to train, then some games."
Players' chief Gordon Taylor had earlier admitted he was relieved Tevez is patching up his relationship with City.
"I'm very glad, having dealt with him and represented him," Taylor said.
"Sorry has been said; sometimes it's the hardest word. Sometimes if you can grasp that nettle early on it makes like so much better for all parties, that's what we're about in football: respect."
Taylor told Sky Sports News: "I hate seeing such top quality players in trouble with their clubs and in trouble with the game.
"This has gone on for too long and it's a real positive step for football, for Carlos Tevez and Manchester City."