Torreira, 24, sustained the injury in the 2-0 victory over Portsmouth in the fifth round of the FA Cup last week.
He left Fratton Park on crutches and the club have said he will miss between eight and 10 weeks.
The Uruguay international will be hopeful of regaining full fitness in time for this year's Copa America and Arsenal boss Mikel Arteta said he hopes to have him back by the end of the season.
"I wouldn't like to rule him out," Arteta told reporters on Monday. "Lucas is really keen and would like to play again this season, whether that is realistic or not we will know in the next two or three weeks.
"It depends how the bone is healing and how he is doing with his rehab."
The club have suffered a number of long-term injuries this season with summer signing Kieran Tierney out with a dislocated shoulder while defender Calum Chambers has not featured since December due to a ruptured knee ligament.
Arsenal could leapfrog rivals Tottenham in the Premier League table with a win away at Manchester City on Wednesday.
Arteta will return to the Etihad for the first time since leaving to become Arsenal's head coach in December.