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On Monday, The Telegraph reported that United had renewed their interest in Milinkovic-Savic, who has been linked with a move to Old Trafford for a number of years.
Milinkovic-Savic has developed into one of the most highly-rated midfielders in Serie A during his seven-year spell at the Rome club.
He has scored 58 goals in 294 appearances for Maurizio Sarri's side since joining from Belgian club Genk in 2015.
Italian outlet Il Messaggero have reported that the 27-year-old, who still has two years remaining on his current deal with Lazio, would be interested in moving to Old Trafford this summer.
United have renewed their interest in the midfielder after failing to make significant progress in their bid to sign Erik ten Hag's number one transfer target Frenkie de Jong from Barcelona.
The Red Devils had a bid, thought to be in the region of £70m, accepted for De Jong but the Netherlands international is reluctant to leave the Nou Camp this summer.
United are also in talks to sign Juventus midfielder Adrian Rabiot as Ten Hag looks to bolster his squad after a disappointing defeat to Brighton in their opening game of the new Premier League season.
The France international has just one year left on his deal in Turin and is reportedly keen to move to Old Trafford this summer.
Sky Sports News claim United have agreed a deal to sign the former PSG midfielder, with personal terms still be discussed.
United are also in talks to sign veteran striker Marko Arnautovic from Bologna after missing out on Red Bull Salzburg forward Benjamin Sesko, who will instead join RB Leipzig next summer.
United have had a £7.2m bid for Arnautovic rejected by Bologna but are expected to return with an improved offer for the former Stoke City and West Ham striker.
However, Bologna director Marco Di Vaio says the Italian club have no plans to sell the Austria international.
"Man United on Arnautovic? We're proud of Man United interest for Marko but we are not planning to sell him," said Di Vaio.
"We want to keep him at Bologna as president said, he's key part of our project – there's not even price tag."