The news means Manchester City will be without their captain for Saturday's FA Cup semi-final against Manchester United.
Mancini had initially said he expected Tevez to be out for a fortnight after he was substituted in the first half at Anfield, but it now appears likely the forward will struggle to return before City's penultimate Premier League game against Stoke on May 14.
"Tevez is not available for Saturday," Mancini said. "I think he will recover in three or four weeks. It is a big problem but we can't take any risks. We have six games left and hopefully we can get Carlos back for the last two or three games."
Balotelli has shown signs of his quality since his summer move from Inter and has scored ten goals in all competitions, but he has also attracted criticism for his work-rate and off-field behaviour. Dzeko, meanwhile, has not yet scored a Premier League goal, although he has scored once in the FA Cup against Notts County and twice in the Europa League against Aris Salonika.
"I still have to decide what to do [in the semi-final], but both players know this is an important chance for them," Mancini said. "Every player knows this is an important moment for the club. If we win, we can go to a final, so every player will give 100%."
In defence, City will at least be boosted by the return of Micah Richards, who was back in training on Wednesday after suffering a hamstring injury last month.
"Micah Richards is available," Mancini said. "He trained two days ago and again yesterday. He will be okay."