Former Chelsea assistant Clarke left his role at Liverpool on Wednesday, where he assistant manager before Kenny Dalglish’s dismissal in May.
West Brom are looking for a successor for Roy Hodgson, who left the club last month to become manager of England.
Birmingham’s Chris Hughton, Claudio Ranieri, Ray Wilkins and German Ralf Rangnick had been linked with the role.
Former Chelsea manager Ranieri, 60, had talks before the Italian was appointed Monaco coach on a two-year contract in May.
Ex-Schalke coach Rangnick, 53, also had discussions with a view to taking over at The Hawthorns, while Hughton has been strongly linked with Norwich followingPaul Lambert’s appointment as Aston Villa manager.
Clarke, 48, parted company with Liverpool on “amicable terms”, the League Managers’ Association (LMA) confirmed on Wednesday.
Last week Liverpool appointed Swansea boss Brendan Rodgers as manager.
Clarke, who was number two to Dalglish, offered his resignation when Dalglish was sacked, but the club rejected it.
He was Dalglish’s first major appointment in January 2011, arriving at Anfield after coaching spells at Newcastle, Chelsea and West Ham.
The LMA said that Clarke, who played over 400 games for Chelsea during an 11-year career with the Londoners, was now “looking forward to the next challenge in his career”.
It was Hodgson’s departure from Liverpool that led to Clarke moving to Anfield alongside Dalglish.
Hodgson steered West Brom to a 10th-placed finish in the Premier League last season.
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.
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.