The defenseman, placed on injured reserve by the Oilers on Wednesday, was injured in the second period of a 6-4 win at the Colorado Avalanche.
"He's going to go back to Edmonton for further evaluation, he's got a hand injury," Hitchcock said. "We'll probably have a further evaluation once we get back to Edmonton, but I would say weeks for sure, and then we'll see from there."
Klefbom, who plays on Edmonton's first defense pair with Adam Larsson, has 15 points (three goals, 12 assists) in 31 games this season.
The No. 19 pick in the 2011 NHL Draft, Klefbom has 109 points (27 goals, 82 assists) in 286 games in six seasons in the League, all with the Oilers.
The Oilers (17-12-2), who have won four straight, visit the Winnipeg Jets at Bell MTS Place on Thursday (8 p.m. ET; TSN3, SNW, NHL.TV).
Edmonton activated forward Drake Caggiula from injured reserve Wednesday; he missed four games with a hand injury.
