The 24-year-old Argentine, who is City's joint top-scorer this season with 10 goals, picked up the injury in City's 3-0 win over Stoke City on New Year's Day and was forced to sit out their FA Cup victory over Watford last weekend.
"Sergio won't be available for at least another week," Mancini told reporters.
"He should make the game against Fulham but we won't risk him against Arsenal. He should be ready by next weekend."
Premier League champions City, who will be able to call on left-back Aleksandar Kolarov after a four-week injury absence, have not won at Arsenal in 37 years.
"We want to end this run but it will be very difficult because Arsenal are a very good team," Mancini said.
"We are missing a number of important players but we can go there and come away with a good result. We are seven points behind United at present but I don't feel we deserve to be so many points behind.
"Our strikers have been coming back into form again and scoring goals and we will need them to keep doing that. Sometimes you go through a period when you don't score so many but we have scored 10 in our last three games so things are improving."
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