Berge, 28, left Burnley for Fulham during the summer of 2024, and has since grown into one of the west London club's most important players.
The Norwegian made 35 league appearances last season, starting 34 of them, and contributed one assist across the campaign.
He holds a contract with Fulham that runs until the summer of 2029, giving the club a strong negotiating position over any potential sale.
Fulham paid £25m to sign Berge from Burnley, and would be looking to sell for a profit should a deal be agreed, with reports suggesting the west London club want any fee to exceed that original outlay.
Manchester United are searching for several midfield additions ahead of next season, with the club's options in that area set to be significantly reduced this summer.
Casemiro is already confirmed to be leaving Old Trafford once his contract expires, creating an additional gap in central midfield that the club will need to fill.
There are also reports linking United with Ederson, who is said to be closing in on a move from Atalanta, as the club looks to reshape its midfield options across multiple positions.
United finished third in the Premier League last season, a campaign that fell short of the club's usual ambitions despite securing Champions League qualification.
Berge's physical profile and ability to operate in a deeper midfield role have made him an increasingly sought-after option for clubs looking to strengthen that area of the pitch.
His form for Fulham over the past two seasons has elevated his reputation considerably since his switch from Burnley, with his performances helping to stabilise a Fulham midfield that has relied heavily on his contribution.