The England left-back limped off the pitch in the second half after picking up the injury in a challenge with Landon Donovan.
Related ArticlesMancini: Bridge committed to playing for EnglandAshley Cole ruled out for three monthsWho can replace Ashley Cole?Nude photos of Ashley Cole lostAshley Cole: highs and lowsD?agrave; vu for EnglandIt means Cole faces a fight to regain his fitness in time for this summer's World Cup, with England opening their campaign in South Africa on June 12 with a match against the United States.
"I'm disappointed because Ashley is a very important player for us, but it can happen that a player gets an injury," Ancelotti said.
"I hope he recovers quickly and comes back before the end of the season. That's a realistic possibility. He will have surgery today and we hope to recuperate him quickly.
"In three months he will be ready to play, so for England he can play in the World Cup."
Manchester City coach Roberto Mancini believes Wayne Bridge is the man to solve England's left-back crisis.
Bridge is said to still be fully committed to his country, despite the recent furore surrounding revelations over John Terry and Bridge's former girlfriend.
Mancini said: 'For England, losing Cole is a problem, but they have good player here in Bridge and he can play in this position in the national team.
'He never told me that he doesn't want to play with Terry. What is important is the national team, but I won't speak to him about it."