Bridge's relationship with former captain John Terry has been shattered following allegations that Terry had an affair with the mother of Bridge's child, Vanessa Perroncel.
Cole limped out of Wednesday's defeat to Everton after a firm but fair challenge from Landon Donovan, captain of the United States team, who will be England's first opponents in South Africa.
"Ashley Cole suffered a fracture to his left ankle in last night's game against Everton and is expected to be out of action for approximately three months," said a club statement.
"We are very disappointed because he's a very important player for us," Chelsea manager Carlo Ancelotti said. "Now we have to do our best to get him recovered quickly, although we have Yury (Zhirkov), who is a fantastic full-back, so we can use him.
"It's difficult to find a full-back like Ashley. Also this year, he's scored three goals and, for us, it's important what he's doing. He's doing important work because I need for our play to have a full-back who pushes up."