The 28-year-old suffered the injury during City's 2-1 win over Hearts on Friday and is now set to miss the start of the Premier League season.
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After scoring 23 goals in all competitions last season, Negredo had been linked with a return to Spain this summer but, earlier in July, the forward insisted that he will remain with the Premier League champions.
"I don't want to leave City," said Negredo.
"I'm very happy and very settled here. I had a difficult end to the season and that's why there has been a lot of talking but I've been very happy here from the first day I arrived."