Santa Cruz hobbled off during City's 2-0 Premier League defeat at Everton on Saturday with a recurrence of the injury that sidelined him over Christmas.
Manager Roberto Mancini had some better news for the eagerly-anticipated clash with United at Eastlands though with midfielder Stephen Ireland expected to return from a hamstring injury that has kept him out for three weeks.
"It looks like a bad injury for Roque, it is to the same calf that was a problem around Christmas. It could be around one month, we don't know exactly when he'll be back yet. It does not require surgery, just treatment," Mancini told City's website on Monday.
Mancini will resist the temptation to hand Patrick Vieira his first start for the club since joining from Inter Milan with the Frenchman still bothered by a calf strain.
"It's better for us and for him if he can be ready to play in another week or 10 days," Mancini said.
"Its better that we don't risk his situation because I would like that when Patrick comes back, he can play to the end of the season. I would want him to be ready to play 20 games, I would prefer him to come back after the next game or for the return match against United."
City are also still without injured defenders Joleon Lescott and Wayne Bridge but Micah Richards, another member of the back four, is fit and relishing the chance to redeem himself against United after his lapse in concentration allowed United to win 4-3 in the Premier League at Old Trafford this season.
"I switched off for a few seconds and the ball was in the back of our net," Richards, who also gave away the penalty which sealed City's first defeat under Mancini at Everton on Saturday, said.
"I got a lot of grief for that from all kinds of people but mistakes happen in football and I hold up my hands. This time I want to be on people's lips for the right reasons."
United manager Alex Ferguson has rotated his squad throughout this year's League Cup but will be tempted to play all his big guns against City, particularly having already gone out of the FA Cup in embarrassing fashion to League One Leeds United.
Whatever team he does send out, Richards said he is expecting a classic match.
"It doesn't get any bigger than United in the cup," he said. "I have played in quite a few derby matches now but you never quite get used to the atmosphere and the intensity and Tuesday's game will probably go up another notch in that respect."
City are aiming to reach their first domestic final since they were beaten by Tottenham Hotspur in the 1981 FA Cup.
