Ashimeru joined Anderlecht on loan from Red Bull Salzburg in the second half of the 2020/21 season, making the move permanent the following summer.
The Ghana international made 39 appearances in all competitions for Anderlecht in the recently concluded season, scoring three times and assisting twice.
The Clarets are interested in signing the central midfielder after choosing his former manager in Kompany to replace long-standing Burnley boss Sean Dyche.
Burnley suffered the drop after losing at home to Newcastle United on the final matchday of the recently concluded season, finishing in 18th place.
Having returned to the Championship for the first time after the 2015/16 season, the Clarets will be looking to make this stint in the second tier as short as possible.
Ashimeru is under contract with Anderlecht until the summer of 2025.
It remains to be seen whether the 24-year-old will be interested in swapping the Belgian first division for the Championship, with his current club set to participate in the playoff round of the second edition of the Europa Conference League as well.
