The former Nottingham Forest winger is due to undergo a medical, with Red Bull Leipzig open to his departure just one year after buying him.
Discussions are still ongoing as to whether the deal will be permanent or simply a loan move.
The West Brom developments are surprising considering that midweek, Crystal Palace were seen as lead contenders for his signature.
This came quickly after Palace appointed Dougie Freedman as sporting director.
Freedman managed Burke at Nottingham Forest and was a huge admirer of the Scotland international.
Burke is unproven at the top level and has not succeeded in Germany, but his performances for Forest in the Championship were enough for Leipzig to spent £13 million on his signature.
Here is a look at how Palace fans reacted on Twitter to the developments...
So Pulis is clearly exacting revenge on Palace. No way it's a coincidence, not Sakho, Winner AND Burke— GazBod (@gazbod) August 24, 2017
What a small, small man.. #CPFC
Ollie Burke could move to West Brom— Jonsta (@Jonsta1990) August 24, 2017
Looking at their midfield, we "need" him mors than they do #cpfc
So, Germans say Burke can't understand complicated tactics/instructions and he rocks up at WBA. Makes perfect sense. #cpfc— Gooders (@RobGooders) August 24, 2017
A lot of talk that it would have been Burke in, Townsend out- maybe we couldn't get a suitable deal agreed for Andros? #cpfc— Jason Baster (@BasterJason) August 24, 2017
We've minced around Burke for days while we try & manoeuvre our chess pieces, meanwhile wba just nip in and do the deal, classic #cpfc— YT (@_IamYT_) August 24, 2017