The 21-year-old has impressed in the Champions League this term, and is a player Chris Wilder is looking at as he seeks to upgrade his squad for the second half of the campaign.
The newly-promoted Blades have exceeded expectations in reaching eighth in the Premier League, and a midfield of Oliver Norwood, John Fleck and John Lundstram have been important to that, but Wilder is determined not to stand still.
West Ham United have also been linked with Berge, who is valued at around £27m.
The deal may not be as advanced as some reports have suggested, but there is definite interest from Bramall Lane ahead of Friday's 11pm deadline.
Genk are thought to prefer a deal which would see Berge loaned back for the rest of the season, but it is thought Sheffield United might seek to bring him in immediately.
The fee, which would include add-ons, would surpass the club record £20m paid for Oli McBurnie in the summer.
So far this month the Blades have bulked out their squad with the arrival of free agent centre-back Jack Rodwell and Nottingham Forest wing-back Jack Robinson.
Berge usually plays as the left of two holding midfielders for the Belgians, and has scored four league goals this season, Genk finished bottom of a Champions League group which included Liverpool and Napoli.
Merseysider Lundstram was on the bench for the recent visit of Manchester City, and has only started three of the last seven matches - perhaps an indication that his energetic game is catching up on him. The Blades only have two more games before their two-week mid-season break.
To Wilder's frustration, Lundstram is one of those players yet to sign the new contract on offer to him, although they will not lose him for free at the end of the campaign, as they will trigger a one-year extension if no new deal has been agreed.
Enda Stevens is another yet to commit to a new contract.
There could also be outgoings before Friday's deadline, and West Bromwich Albion are showing an interest in taking striker Callum Robinson on loan for the second half of the season. The summer signing from Preston North End has been lightly used since the autumn.
It would be a surprise if Wilder were to weaken his attacking options, but he has been linked with a surprise move for Richairo Zivkovic, a Dutch Under-21 international currently playing in China.