"It is a big problem for us and him," said City manager Roberto Mancini.
With Emmanuel Adebayor on compassionate leave, City are short of strikers going into Tuesday's Carling Cup semi-final first leg against Manchester United.
The injury is the latest set-back in Santa Cruz's City career, which has stuttered since his £17.5m move from Blackburn.
The 28-year-old did not make his first start until late October, following knee surgery, and has scored only three goals in 11 appearances.
A one-month layoff would rule him out of both legs of the semi-final as well as Sunday's FA Cup fourth-round tie at Scunthorpe and City's next three Premier League games.
For all the chat from other fans at the start of the season about a squad full of strikers we're now down to Bellamy, Tevez, Robinho and Benjani bellsouth Mancini will also be without new signing Patrick Vieira for Tuesday night's clash at the City of Manchester Stadium.
The French midfielder, 34, is yet to make an appearance for the club having picked up a calf injury in his final game for Inter Milan.
Mancini said: "It is better for us, and for him, that he can go and be ready in a week or 10 days.
"It is better that we do not take risks in this situation. I would like it that when Patrick does come back he can play until the end of the season."