The 28-year-old is heading back to the Premier League club that he has not played for since September after defying Italian Mancini's instruction to warm up during the Champions League defeat at Bayern Munich.
Speaking for the first time since returning home to Argentina without City's permission in November, Tevez gave his version of what happened in Germany.
He said Mancini had been arguing with striker Edin Dzeko who had just been substituted when the manager approached Tevez.
"He saw me on the bench and as he was angry, he sent me again like a dog to warm up. Because he said it to me in a bad tone I refused," Tevez said in an interview with Fox Sports broadcast on Monday.
Mancini said at the time Tevez was "finished" at the league leaders but has since opened the door for a return by hinting the striker could still play a part in their pursuit of a first league title since 1968.
Tevez said he would gladly play for the club again if Mancini was serious about having him back, and was aware he would have to pull out all the stops to win over fans who have burned their former captain's shirts in anger at his behaviour.
"If I was wrong, I say sorry. I sincerely believe I did not make a mistake," he added.
"If it's true (Mancini would welcome me back), of course I like that, if it's (meant) for the media, no."
An adviser to Tevez's representative Kia Joorabchian, Paul McCarthy, told Reuters the player was returning to England and could be playing for City in the next two to three weeks.
"He is due back in England on Tuesday and hopefully will start training this week," McCarthy said. "He has not just been playing golf in Argentina, he has done a lot of fitness work too but is probably still two or three weeks away from being match-fit."
Mancini strongly hinted on Sunday that Tevez could still play a role for City.
"Everyone knows Carlos is a top player. If he was here and playing it would be better because Carlos can change games," Mancini said.
