The 30-year-old has been with Napoli since 2019, representing the club on 225 occasions in all competitions, scoring 15 goals and registering 35 assists in the process.
The club captain has again been a vital player this term, scoring twice and registering seven appearances in 41 appearances, and he signed a new five-year contract with his Italian side last August.
However, according to Il Mattino, the Italy international is considering seeking a move away during this summer's transfer window.
The report claims that Man United and Villa are both attentive to the situation, with the English pair potentially battling for his services when the summer transfer market opens for business.
Man United, Villa could battle for Di Lorenzo
Di Lorenzo played a key role in Napoli winning the 2022-23 Serie A title, but a tough campaign for the Blues has left down down in seventh position in Italy's top flight.
Napoli are only two points behind sixth-placed Atalanta BC in the battle to secure European action for next season, but they are some 10 points off fourth-placed Bologna, with the club therefore unlikely to operate in the Champions League next season.
Man United, under normal circumstances, would expect to beat Villa in a battle for a player, but the Red Devils are also facing the prospect of missing out on Champions League football for next term.
The 20-time English champions are sixth in the Premier League table, 11 points behind fifth-placed Villa, albeit with a game in hand on Unai Emery's team.
Is right-back a priority position for Man United?
Man United have two senior right-backs at the club in shape of Diogo Dalot and Aaron Wan-Bissaka, and the former has been one of the club's most consistent players this season.
Wan-Bissaka has also been a regular despite some fitness issues, making 21 appearances in all competitions, but he is currently being used as a left-back due to the team's problems in that area of the field.
Luke Shaw and Tyrell Malacia are both unavailable for selection, while another stand-in option at left-back, Victor Lindelof, is also on the sidelines with a hamstring problem.
Man United activated a 12-month extension in Wan-Bissaka's contract earlier this year to take him until June 2025, with the club protecting his valuation.
The 26-year-old could be on the move this summer, though, with former club Crystal Palace said to be interested, and his departure would place Erik ten Hag's side in the market for a right-back to both support and provide competition for Dalot moving forward.