Claudio Yacob, Steven Reid and Jonas Olsson have all been forced to miss training at some stage this week.
Shane Long is expected to be ready to face the Swans at The Hawthorns, but Chris Brunt is still struggling with an ankle complaint.
The Baggies skipper sat out a 1-0 defeat to Chelsea last weekend as a result of the knock and Clarke admits he faces a race against time to prove his fitness.
He said in the Birmingham Mail: "We thought Chris would train yesterday (Tuesday) but he trained on Monday and still felt something so, fingers crossed, he'll train later this week.
"He and Shane twisted their ankles in the game against Sunderland.
"Shane was still feeling it a bit on Saturday, but he's okay. Claudio has a knock, hasn't trained, but should be okay. Steven Reid has a knock, hasn't trained, but he should also be okay.
"Jonas didn't train on Monday but trained yesterday and Youssouf Mulumbu had a migraine but he's fine now. Hopefully everyone will be fine tomorrow."