The West Brom captain has already been ruled out of Saturday's trip to Southampton due to the injury he picked up in the 1-1 draw with Newcastle last weekend.
It leaves the West Brom boss short of numbers for the meeting with Mauricio Pochettino's in-form side as they look to earn their first win since early March.
Clarke said: "Chris' injury is a little bit of a concern for us.
"It would be nice to think he'll only miss one game but there is a possibility he will miss more than one game.
"He's not going to make it this week and it is fingers crossed for next week. Hopefully the medical department will do the job they've done all season and get him back quickly."
Clarke's problems have been exacerbated by another injury to defender Billy Jones and the suspension of Youssouf Mulumbu.
"It puts us down to the bare bones," he continued. "But Youssouf will be available again next week.
"Billy picked up a really nasty gash on his leg and is doubtful. But I would be hopeful Billy will maybe make it."
There was some good news, however, with the club making moves to secure the futures of several key players with Mulumbu, Gareth McAuley and Shane Long having all held talks over extending their contracts.
Clarke is hopeful new terms will be agreed, and picked out McAuley for special praise.
He said: "I would like to think Gareth will be here next year, he has been our best player this season, consistently the best.
"I am sure he will pick up all our player of the year awards and I would like to think, as I sit here now, Gareth McAuley will still be with the Albion next season.
"I want him to get the deal he deserves and I would like it put to bed as soon as possible but it is not my department. I leave that to other people."