Everton and Newcastle also want Sinclair but are yet to make a formal bid for the winger, who made only 11 appearances for Manchester City during the last campaign.
Clarke is determined to add to his squad before the end of the transfer window and feels a new signing would give his side an extra boost prior to the big kick-off.
"We've made an enquiry to Manchester City," he told a regional newspaper.
"By and large the targets we've identified we haven't been able to persuade to come so far. Some of them have maybe been too expensive."
Sinclair was left out of the City squad for their final pre-season match where they suffered a 3-1 defeat at the hands of Premier League rivals Arsenal in Helsinki.
City boss Manuel Pellegrini expects the future of the former Chelsea man to be sorted by the end of this week.
The transfer to West Brom will bring to an end an unsuccessful year in Manchester for Sinclair, who made only two league starts since his £5.2m move from Swansea last summer.