The Baggies hotseat was vacated by Roy Hodgson at the end of last season in order to take the England job.
Albion have moved quickly to sound out Clarke after he left Anfield and sources close to the club believe he could even be appointed by the weekend.
Former Schalke boss Ralf Rangnick had looked set to take charge at the Hawthorns, but he appears to have now dropped out of the running to succeed Hodgson.
Clarke has gained plenty of experience working as number two at Chelsea, West Ham and most recently at Liverpool under Kenny Dalglish.
But the 48-year-old is believed to be keen to step into a managerial role and Albion fit the bill.
Clarke will have a hard act to follow in Hodgson, who guided West Brom to a top-10 finish in the Barclays Premier League last season.
It was Hodgson's departure from Liverpool that led to Clarke moving to Anfield alongside Dalglish. Now he has the chance to again replace Hodgson, who is in charge of England's Euro 2012 campaign.
Among his first tasks will be to try and make goalkeeper Ben Foster's loan move from Birmingham a permanent one, although he is also a target for QPR.
Key central defender Jonas Olsson - currently part of Sweden's Euro 2012 squad - also has only a year left of his current deal.
Former Scotland international Clarke worked under Ruud Gullit and Jose Mourinho at Chelsea and Gianfranco Zola at West Ham.