The 25-year-old sustained the injury in the second half of Tuesday's 1-0 Championship home defeat to Derby and was forced off early in the second half.
Boss Stuart Gray confirmed it was not a tear, but will wait for scan results to find out the full extent of the problem, which is likely to keep Coke out of Saturday's Yorkshire derby at Huddersfield.
"It's not a tear, but it's a hamstring problem," Gray said. "It was bad enough for him to come off because when Kieran lies on the pitch, you know he has got a problem. It's a scan job and I wouldn't want to put a timescale on how long he will be out for."
The injury is another in a long line for Wednesday, whose treatment room has been overflowing this season.
It comes a week after the Owls lost midfielder Jose Semedo for up to eight weeks with a broken leg and Gray admits it has been a frustrating time.
"At the end of the day, you look what you are doing in training, but I just think it's a bit of a freak season for us," he added.
"At one point, we had five centre-halves in the treatment room, so it is very, very frustrating.
"The most frustrating thing is the central midfield pairing of Kieran Lee and Jose Semedo was doing so well, and suddenly I've lost them both.
"But that is part and parcel of football and it opens the door for someone else to stake a claim for a place."