The Red Devils have been in the market for a new central-midfielder all summer, the expectation having initially been that Casemiro would be moved on and replaced.
However, the Brazilian has remained at Old Trafford and starred in the club's opening fixtures against Manchester City and Fulham in the Community Shield and Premier League respectively.
Erik ten Hag still wants to add to his engine room, though, with Sofyan Amrabat having returned to Fiorentina and offers being made for Scott McTominay.
Berge is viewed as an attainable target after relegation with Burnley, who are happy to cash in on the Norway international for the right price.
Nevertheless, it appears that United will now need to react to the latest development in the race to sign the 26-year-old.
Fulham steal march on United
According to Sky Sports News, Fulham have managed to negotiate a fee with their Burnley counterparts over the past 24 hours.
The report claims that the Cottagers are willing to pay in the region of £20m, plus add-ons, to secure Berge's signature.
Fulham have money to spend following Joao Palhinha's switch to Bayern Munich, as well as needing to react to losing 1-0 to United on Friday despite posing a constant threat at Old Trafford.
Berge is now in line to undergo a medical with Fulham on Tuesday, it seemingly yet to be determined whether he may be prepared to wait for United to make a decision on their next move.
United are said to have been in open dialogue with Burnley, although it is unclear whether they will now look to match or exceed Fulham's offer.
Since joining Sheffield United in 2020, Berge has spent the last four years in the North of England, that potentially handing an advantage to United if they did wish to improve their bid.
Will United be prepared to react?
There has previously been talk of Fulham holding an interest in McTominay, yet nothing will come of that if Marco Silva's side beat United to Berge.
The report adds that Ten Hag and club officials will only give the green light to an exit for the Scotland international if he specifically requests an exit.
Although he has less than 12 months remaining on his contract, United are in a position where they can extend the academy graduate's terms by a further 12 months.