Reports suggest the deal would be worth around £60m.
The Brazilian international, who spent the 2014-15 season on loan at Porto, has been at Real since 2013, winning three LaLiga titles and five Champions Leagues with 336 appearances across all competitions.
Man United sit bottom of the Premier League table after their 2-1 opening home defeat to Brighton was followed up with a 4-0 loss at Brentford in a disastrous start to Erik ten Hag’s reign as manager .