Albion's £15million signing is yet to start a game for the Baggies since joining from Bundesliga club RB Leipzig in the summer.
Head coach Tony Pulis is eager to get the pacy 20-year-old into his squad, but Burke has struggled to stay fit so far.
"He's done his hamstring which is really disappointing," said Pulis. "He played for Scotland (U21s) and did really well.
"He would have been a great player to have on the bench (against Leicester), especially when it got really stretched, because his pace is frightening and he wants to run in behind.
"He's been disappointing in respect of his injury, not his personality, he wants to play.
"It's unfortunate that the kid's tweaked his hamstring and we can't get it right at the moment."
Burke has only made two substitute appearances for the first team so far, and hasn't featured since a half-hour cameo away at Brighton.
He returned from his hamstring injury just before the international break, and featured for Albion's under-21s in the Checkatrade Trophy before captaining Scotland's U21s.
But after suffering a set-back to his injury, he tweeted: "Frustrating times but hope to be back out and playing soon."
Albion bought the Nottingham Forest academy product from Leipzig in a deal worth a total of £15m.
Only Nacer Chadli has cost the Baggies more, and the Belgian returned from his own injury in stunning fashion on Monday night with a free-kick goal against Leicester.
Now Pulis wants Burke back in contention too.
"I have something in my mind that I think will suit everybody," he said. "It's waiting for Burke and Chadli to come back and try and fit them in.
"We're pleased to have Nacer back in the team, having Nacer and (Kieran) Gibbs on that left hand side will give us a little bit of quality that we need."