The Baggies were rocked last week when Dara O'Shea was ruled out for between four and six months after injuring his ankle whilst on international duty.
The severity of that injury saw Albion swoop to sign free agent Bryan on a two-year-deal.
Having not had a proper pre-season, the former Sheffield United man won't be ready to face Millwall tomorrow.
But boss Valerien Ismael hasn't ruled out the 24-year-old featuring against Derby on Tuesday.
And the Frenchman also revealed Clarke is training well, have initially been expected to be out for another four weeks.
"Matt Clarke is out (of Millwall) but there are good signs he might come back earlier," Ismael said.
"We don’t want to rush him, but we see very good performances from him in training, the sessions he’s made so far.
"We’ll see in the next seven or eight days."
On the availability of Bryan, Ismael said: "We’ll see now day by day, but we have to build him up, we have a plan for him.
"Our sports science guys have done a great job so far.
"For sure it’ll be too soon for Millwall, that is clear.
"But we will see if it’s a case for Tuesday or for next Saturday."
Ismael believes it was important Albion acted quickly to sign Bryan following the injury sustained by O'Shea.
"When something happens, especially the day after the deadline, to go on the free agent market is never easy," the Frenchman said.
"You know the player is two months behind – there is no pre-season, especially with our way to play.
"We needed to be really clinical and to make sure that we got that feeling that one of the guys could be a good case for us.
"I think with Kean’s profile – 24 years old, left footed, young, hungry, we heard a lot of positive feedback from former coaches and players.
"He gave us the same feeling that he has the same profile as Dara.
"There is a lot of speed, aggression for the position, on the front foot all the time.
"It made sense for us to do the deal and we’ve got a good feeling.
"Now our job is to make sure we have a good build up with him, not to rush, but at the same time we will need him so we need to find the right balance – when is the right time for him to play and to make sure he can compete."