Tevez sustained a hamstring injury during the 3-0 defeat to Liverpool last month but was initially expected to return in time to face Stoke at Wembley on May 14.
However, City now have only two games to restore Tevez's match fitness and Mancini has warned he will have to play against Everton on Saturday or Tottenham on Tuesday if he is to be part of the FA Cup final side.
"I think Carlos needs to train because it's impossible to play without training," he said. "We have ten days. We are waiting."
He added: "It is very difficult for him to play against Everton and, if he doesn't play against Tottenham, it will be impossible for Carlos to play in the final."
Micah Richards, though, is making good progress from his hamstring injury and Mancini said the defender is "very close" to a return and could be on the bench at Goodison Park this weekend.