The Red Devils, having appointed Dan Ashworth as their new sporting director, are expected to be active during the summer transfer window despite financial restrictions.
According to the Daily Mail, a deal for Matthijs de Ligt is advancing, with the 20-time English champions stepping up their interest in the Bayern Munich centre-back.
The report claims that the Netherlands international has signalled that he is open to moving to Old Trafford, which would see him reunite with former Ajax manager Erik ten Hag.
Man United are believed to be willing to hand De Ligt a five-year contract, with the package for the 24-year-old potentially costing in the region of £42m, including add-ons.
Man United 'making progress' on De Ligt deal
However, the Red Devils are still said to be keen to sign Everton's Jarrad Branthwaite, with the Englishman long thought to be regarded as their leading centre-back target.
The club are already believed to have had an offer in the region of £45m turned down by Everton, with the Toffees thought to value the defender in the region of £80m.
For a deal to become a possibility, Everton would have to lower their asking price, but Man United are allegedly open to bring in two new centre-backs this summer.
Raphael Varane has left on a free transfer, while Victor Lindelof and Harry Maguire are being linked with moves away; Jonny Evans's future is also up in the air, with the Northern Ireland international yet to agree on a new deal.
Zirkzee, Ugarte 'wanted' by Man United
Man United are also said to be pressing ahead with a move for Bologna's Joshua Zirkzee, with the Dutchman said to be the club's leading target in the final third of the field.
Zirkzee, who has a £34m buy-out clause in his contract, has scored 14 goals and registered nine assists in 58 appearances for his current side in all competitions.
The Red Devils want to bring in competition for Rasmus Hojlund this summer, with Anthony Martial's departure on a free transfer leaving the club with just one senior centre-forward.
A new central midfielder is also thought to be a priority amid claims that Christian Eriksen and Casemiro could both be on their way out of the club this summer.
Paris Saint-Germain's Manuel Ugarte is said to be a leading target, with the Uruguay international potentially available for around £50m during the summer transfer window.