Pep Guardiola's side are likely to be waving goodbye to long-serving midfielder Fernandinho, whose contract expires at the end of the season, and City are expected to bolster their midfield if the 35-year-old does leave.
Berge has not featured for Sheffield United since December due to a serious hamstring tear - which he has undergone surgery on - but the Norway international impressed for the Blades before entering the treatment room.
According to Belgian outlet Voetbal24, City will look to strike a deal for Berge if Fernandinho's departure is confirmed, but they are not the only Premier League side interested in his services.
The report adds that Liverpool will consider making a move for the Blades midfielder as Georginio Wijnaldum runs the clock down on his Anfield deal, but Sheffield United are not prepared to sanction a cheap deal despite their seemingly unavoidable relegation to the Championship, as his former club Genk will pocket 20% of any future transfer fee.
Berge is under contract at Bramall Lane until 2024, and he has registered two goals and two assists from 30 appearances for the club since arriving in January 2020.