Clarke pulled up in the 2-1 win at Blackburn with a hamstring issue and boss Valerien Ismael, whose side host Arsenal in the Carabao Cup tomorrow night, has confirmed he will miss the next several games.
Two unnamed Albion players have also tested positive for Covid-19 and are now in isolation.
On Clarke, Ismael said: "We received yesterday the information that he will miss a lot of games – five to six weeks.
"We need to deal with the situation. It's bad news for us, but it is what it is.
"You can't influence injuries or Covid. They're two situations we have to deal with.
"Now we have first case of injury, so we need to make sure he recovers well and comes back stronger.
"It's a good thing that the season is very long in the Championship, so we'll see him soon."
Ismael was unable to name the two players the Baggies will be missing through Covid but said: "We have two Covid cases with players. So, we need to make sure everything is in the right way.
"They need to isolate. It's a medical case so I can't give the information.
"Covid is always a worry at the minute. You cannot control this.
"You can only get the jab and make sure you do the right thing, but we knew from the beginning we had to deal with injuries and Covid.
"We'll deal with the situation and next week is an international break, so we'll have more time to get the players back."
Albion, meanwhile, remain 'calm' as they look to add to the squad before the window shuts next Tuesday.
A striker is on their list, while early discussions with goalkeeper Sam Johnstone over a new contract are ongoing.
"We know behind the scenes what we have to do. We try and we have some more days," said Ismael on potential signings.
"We are calm. We don't need to rush as we know exactly what we want.
"My expectation is that we can do something until the end.
"You never know because it's always a difficult situation, but behind the scenes Ken (Xu Ke, chief executive) and Ian Pearce (head of recruitment) have done a great job so far.
"We try to do everything to make sure we can get the right players."
Ismael added on the Johnstone situation: "Progress is a big word but we keep talking.
"I don't think we can get a result quickly, but at least all the parties are talking.
"We have a good feeling, but we need to be patient on that."