Clarke is the new favourite to replace England manager Roy Hodgson in the Baggies hotseat.
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.
The 48-year-old is believed to be keen to step into a managerial role and Albion fit the bill.