The 27-year-old is said to be ready to sign on the dotted line before heading off to the World Cup, making him the club's second-highest earner behind January signing Alexis Sanchez.
Real Madrid's rumoured interest in the Spain international has refused to go away, but The Sun suggests that years of speculation will end in the coming weeks as both parties are happy with the contract offer.
A United insider is quoted by the publication as saying: "The club want it sorted. It has to be done in their eyes. They want David can go to the World Cup and come back knowing everything is settled.
"Zlatan [Ibrahimovic] leaving has freed up more wiggle room on wages and United don't want to drag it out. They want to show the world that David won't be leaving for anywhere. It's a real statement mark for them."
De Gea has kept a total of 91 Premier League clean sheets for the Red Devils in 231 appearances since arriving from Atletico Madrid nearly seven years ago.