Sane had to come off at halftime following a challenge from Cardiff midfielder Joe Bennett, who was booked for the challenge before later being sent off for another foul.
"Sane has since undergone scans that confirm the 22-year-old is set for a spell on the sidelines - although no timeline has yet been set for his return," league leaders City said on their official website (www.mancity.com).
City manager Pep Guardiola was angered by Cardiff's physical approach and said after Sunday's game that Sane "will be out for a while, minimum two to three weeks or a month".
"I have said many times, all the referees have to do is protect the players - not just the Manchester City players, all players," added the Spaniard.
Sane has been on top form this season with 11 goals and 14 assists in all competitions as City compete for four trophies.
(Writing by Ken Ferris; Editing by Pritha Sarkar)
