The governing body had sanctioned a rematch between Wilder and Tyson Fury but the Briton will take on another bout before potentially facing the American again.
The WBC also ordered Breazeale to face Dillian Whyte to determine Wilder's mandatory challenger last month.
But it has now cancelled that fight.
Wilder, 33, and compatriot Breazeale's camps have until 4 April to agree a deal.
Breazeale, 33, has lost one of his 21 fights - a seventh-round knockout by world heavyweight champion Anthony Joshua in 2016.
Britain's Joshua, 29, will defend his IBF, WBA and WBO titles against unbeaten American Jarrell Miller, 30, in New York on 1 June.
A rematch between Wilder and Fury had been anticipated following their thrilling draw in Los Angeles in December.
However, Fury signed a lucrative deal last month with US broadcaster ESPN, which promoter Frank Warren said means he must take a warm-up fight before facing Wilder, who is aligned with the Showtime network.