The injury will rule Aguero, who has scored 26 goals this season, out of the first leg of City's Champions League last-16 tie against Barcelona as well as Premier League and FA Cup games against Chelsea.
The game at White Hart Lane had been Aguero's fifth since returning from a month-long lay-off with a calf injury.
Manuel Pellegrini: "The doctor is seeing him - maybe one month, but we are not sure absolutely.
"It is very difficult. Sergio is a top player for us but I think we can replace him.
"As with the other injury, we hope we can continue to do it.
"But, of course, he is a very important player for us.
"Sergio is just coming back from an injury - one month that he couldn't play.
"All the other strikers, Alvaro Negredo, Edin Dzeko and now Stevan Jovetic, replace him."
Saturday February 1
12:45 Newcastle United v Sunderland
12:45 West Ham United v Swansea City
15:00 Cardiff City v Norwich City
15:00 Everton v Aston Villa
15:00 Fulham v Southampton
15:00 Hull City v Tottenham
15:00 Stoke City v Man Utd
Sunday February 2
13:30 West Brom v Liverpool
16:00 Arsenal v Crystal Palace
Monday February 3
20:00 Manchester City v Chelsea
1 Man City 53
2 Arsenal 52
3 Chelsea 50
4 Liverpool 46
5 Tottenham 43
6 Everton 42
7 Man Utd 40
8 Newcastle 37
9 Southampton 32
10 A Villa 27
11 Swansea 24
12 Norwich 24
13 Hull City 23
14 C Palace 23
15 West Brom 22
16 Stoke City 22
17 Sunderland 21
18 West Ham 19
19 Fulham 19
20 Cardiff 18