Coughlin had led Berwick to the Third Division title and promotion last season.
But after a disastrous start to their 2007/08 Division Two campaign he stepped down from the post and Renwick took the reins.
"An announcement will be made this week regarding a new manager," the statement continued.
General manager Jimmy Crease will take over as caretaker manager of the Wee Rangers for the second time this season, with the assistance of Ian Little.
Berwick ended their last home game in Division Two at the weekend to a chorus of boos from their fans following a 5-2 drubbing by Raith Rovers.