The 27-year-old Argentine international injured his hamstring in the 3-0 defeat at Liverpool on Monday and, after initial hopes he could make a quick recovery, is now likely to be sidelined for much of the Premier League run-in.
"Tevez is not available for Saturday. I think he will recover in three or four weeks. It is a big problem but we cannot take any risks," Mancini told a news briefing.
"We have six games left and hopefully we can get Carlos back for the last two or three games."
That could mean the striker has a chance of making the final against Stoke City or Bolton Wanderers on May 14, if City get past United at Wembley on Saturday.
Tevez has played a major part in City's assault on a Champions League place with 19 league goals this season, and although they are fourth, Mancini's side are only three points ahead of Tottenham Hotspur who have a game in hand.
Without the Argentine, City will have to put their trust in Edin Dzeko and possibly Mario Balotelli to lead the attack, although Mancini said he had still to make his mind up.
"I still have to decide what to do," he said. "But both players know this is an important chance for them."
The City manager was able to report better news on full-back Micah Richards who has returned to training after a hamstring injury with Mancini saying he is available for Saturday.
