Bookmakers slashed the odds on the German taking charge at The Hawthorns following reports that the former Werder Bremen boss held talks with Albion's representatives in London on Tuesday.
Ex-Valencia boss Quique Flores was also a front-runner for the vacancy but he has since drifted in the betting as a compensation package would have to be paid to United Arab Emirates club Al Ain.
Gianfranco Zola, out of work since leaving Watford last month, also re-emerged as a possible candidate but the Italian is thought to be no longer under consideration following Tuesday's negotiations.
Schaaf is a one-club man after spending 17 years as a player and 14 years as a manager of the Bundesliga club before leaving at the end of last season.
The 52-year-old won six domestic trophies with Bremen, including the Bundesliga crown in 2004, and led the club to the UEFA Cup final in 2009, but lost to Shakhtar Donestk in extra-time.
Clarke was sacked by Albion on December 14 after a run of four straight defeats left the Baggies just two points above the drop zone.