Berge made 109 competitive appearances during a three-and-a-half-year stay with Sheffield United, before he completed a move to completed last summer.
The Norway international was a regular fixture in the Burnley squad last season, featuring in 37 of the club's 38 Premier League matches.
However, he was unable to help Burnley stay in the Premier League and is now likely to leave the club before the transfer deadline.
Fulham rivalling Man United for Berge
A recent report claimed that Man United are interested in Berge and have stepped up their pursuit with their first official offer.
However, Man United may not have a clear path to Berge, with Fulham also looking to sign the Norway international this summer.
According to BBC Sport, the Cottagers have submitted an initial £20m bid for Berge plus a further £5m in add-ons.
Fulham are seemingly keen to make Berge their fourth signing of the transfer window, having already recruited Emile Smith Rowe, Jorge Cuenca and Ryan Sessegnon.
Fulham keen to strengthen midfield
While they appear to be competing with the Red Devils, Fulham have also been heavily linked with one of Manchester United's midfielders.
The Cottagers have failed in their attempts to strike a deal for the Scotland international, but they are reportedly prepared to submit a fresh proposal.
Fulham head coach Marco Silva is hoping to bolster his midfield options after seeing Joao Palhinha leave the West London club earlier this summer.
The Cottagers sold Palhinha to German giants Bayern Munich for a club-record fee of £47.5m, leaving Silva with Harrison Reed, Sasa Lukic, Tom Cairney and Andreas Pereira as his main central midfield options.