The centre-back moved to Atalanta's academy from Brescia in 2015, progressing through the various age groups at the Italian club before debuting for the first team in 2021.
Scalvini, 20, has already made 82 senior appearances for his current side, scoring four goals and registering four assists in the process.
The Italy international has also again been an important player for Atalanta this season, scoring once and providing two assists in 27 appearances in all competitions.
Scalvini has a contract with La Dea until June 2028, but there has been speculation surrounding his future over the last 12 months due to his impressive performances.
According to Calciomercato, Man United have been in touch with Atalanta to discuss a move, and the defender's asking price is believed to be around €39.7m (£34m).
The report claims that the English club's main competition this summer is likely to come from Borussia Dortmund, who have also contacted the Italian team.
Man United signed Rasmus Hojlund from Atalanta last year, though, while they also negotiated a deal for Amad Diallo from the Italian outfit in January 2021.
There is believed to be a solid relationship between the two clubs, which could give the 20-time English champions the edge when it comes to any transfer talks.
Man United are believed to be planning to move for two young centre-backs during this summer's transfer window as part of the rebuild under incoming minority shareholder Sir Jim Ratcliffe.
Everton's Jarrad Branthwaite is thought to be a leading target, but the Englishman is also wanted by the likes of Tottenham Hotspur and Real Madrid.
Raphael Varane is out of contract at the end of June, with the club facing a big decision when it comes to the ex-France international's future.
Jonny Evans also only has a deal until the end of the season, while Victor Lindelof could be sold, but Harry Maguire's form could lead to him being kept for the 2024-25 campaign.
Meanwhile, Lisandro Martinez, who is currently on the sidelines recovering from a knee injury, is seen as a vital part of the club's plans moving forward.