Rumors were swirling about the Norway international potentially joining Manchester United.
Meanwhile, Fulham were said to have an interest in United midfielder Scott McTominay.
However, the BBC states they have now put in a bid of £20 million plus add-ons for Berge.
The Cottagers want to sign a midfielder to fill the void left by Joao Palhinha, who signed for Bayern Munich in July.
Crystal Palace defender Joachim Andersen is also a target, but that deal is not yet done.