Branthwaite, 21, has been one of Everton's best performers during a tough campaign, and his form at Goodison Park is believed to have attracted attention from a number of leading European clubs.
Man United are expected to move for a couple of centre-backs this summer, as Raphael Varane and Jonny Evans are both out of contract at the end of June, while Victor Lindelof could potentially depart for the right price.
Branthwaite is thought to be the club's leading target, although it is believed that Everton value him in the region of £70m.
According to the Daily Star, Man United are hoping to reduce the asking price by including Maguire as part of a swap deal, with the ex-Sheffield United youngster's future also the subject of speculation.
Man United 'keen to sign Branthwaite this summer'The report claims that Maguire's involvement could see the package reduced by £20m, but it remains to be seen whether the centre-back is prepared to leave Old Trafford.
West Ham United had an offer accepted for the England international last summer, but he refused to leave the Red Devils and has managed to regain his spot in the side this term.
Maguire has been vital for Man United of late due to injuries for Varane, Evans, Lindelof and Lisandro Martinez.
The 31-year-old has made 31 appearances for the 20-time English champions during the 2023-24 campaign, scoring four goals and registering two assists in the process.
Maguire's contract is due to expire next summer, although the club do have the option of activating a further 12 months to take him until June 2026.
Branthwai Read on sportsmole.co.uk