It was confirmed earlier this week that the 32-year-old suffered ligament damage in the 1-0 defeat to Fulham last Saturday, and Adams concedes that it is a big blow for his side to lose the influential playmaker.
"Wes went to see the specialist on Wednesday and he's going to be out for a number of weeks," the Canaries boss told the club's official website.
"It's ankle ligament damage, as we first feared. It's a shame, obviously you keep your fingers crossed, but it's going to be number of weeks before he's back with us unfortunately."
Norwich meet Sheffield Wednesday at Hillsborough on Saturday.