Marco Silva’s side have agreed to pay an initial £20m, with a further £5m in add-ons.
Berge, 26, would be Fulham’s fourth signing of the transfer window, following the arrivals of Emile Smith Rowe, Jorge Cuenca and Ryan Sessegnon.
Fulham are looking to increase their midfield options after Joao Palhinha signed for Bayern Munich last month.
The Cottagers have also tried to bring in Crystal Palace defender Joachim Andersen, with their last bid – £25m plus £5m in add-ons – falling short of Palace’s valuation.
Berge, reportedly a target for Manchester United, has not featured in Burnley’s first two games of the Championship season, with boss Scott Parker citing an injury as the reason.
He played 37 Premier League games last season as the Clarets were relegated from the top flight.
He scored his only Premier League goal of the campaign in Burnley’s 2-0 win at Craven Cottage in December.