The Hammers suffered a 2-0 defeat to the newly-crowned Bundesliga champions in last week's first leg held in Germany, courtesy of two late goals from substitutes Jonas Hofmann and Victor Boniface.
Goalkeeper Lukasz Fabianski made an impressive nine saves to ensure David Moyes's side remain in the tie following a powderpuff attacking display - registering just one shot on goal and boasting only 27% possession throughout the 90 minutes.
West Ham sorely missed the presence of top scorer Jarrod Bowen, who was ruled out of the first leg and also missed last weekend's 2-0 Premier League defeat at home to Fulham due to a back issue.
The Hammers also had to cope without Alphonse Areola (groin) and Kalvin Phillips (hamstring) due to injury, while Edson Alvarez was absent against Leverkusen through suspension.
Bowen, Areola, Phillips all train ahead of Leverkusen clash
West Ham can welcome back Alvarez on Thursday and three more players could also return as Moyes has confirmed that each of Bowen, Areola and Phillips have taken part in first-team training.
Providing a fitness update on his squad at a press conference on Wednesday, Moyes said: "It was great to see [Bowen] out there, it was the first time we've had him out (since his injury).
"Not sure yet [if he will be involved], we had Kalvin [Phillips] out as well but I'm not sure about him either. Alphonse [Areola] was out as well but I'm not sure about him either. They'll all be pretty late calls tomorrow."
"Most people know exactly what Jarrod's done for us over this season, and other seasons also," Moyes added. "If he came back it would give us a real lift as well and give the crowd a lift.
"But overall, we've lost Lucas [Paqueta] through suspension and we've had Jarrod out with injury so we have to find a way of making some magic happen. We have to try and do that tomorrow."
Paqueta, Emerson ruled out, Mavropanos a doubt
Two West Ham players who will miss Thursday's visit of Leverkusen are Paqueta and Emerson Palmieri; the Hammers duo picked up their third yellow cards in the Europa League in last week's first leg and are therefore suspended.
Moyes could also be without defender Konstantinos Mavropanos who has emerged as an injury doubt after missing first-team training on Wednesday.
After sustaining an unspecified injury in the first leg against Leverkusen, Mavropanos picked up a knock to his ankle in the defeat to Fulham on Sunday and was forced off on the hour mark.
The potential absence of the Greece international will likely see captain Kurt Zouma recalled to partner Nayef Aguerd in the heart of the defence.
Meanwhile, Moyes is set to weigh up how to deploy Aaron Cresswell in his starting lineup; the Hammers boss used a back three to match the formation of Leverkusen last week, with Cresswell operating as a left-sided centre-back, but the absence of Emerson could see him operate as either a left wing-back or left-back.
Left-footed defender Angelo Ogbonna is another player available to start in the back three, but if Cresswell retains his spot as a centre-back, then Ben Johnson could be tasked with filling in as a left wing-back.