The Independent says United officials will open talks with De Gea over a new contract in the coming weeks, with manager Jose Mourinho keen to tie the Spain international to a long-term contract by the start of the new year.
The 26-year-old, who joined United from Atletico Madrid for £18.9m ($25.6m) in 2011, has two years left on his current deal.
Real Madrid have been persistently linked with De Gea and came close to signing the Spain international in 2015. However, the transfer collapsed on deadline day as the necessary paperwork was not filed on time.
United's confidence in extending De Gea's contract is reported to stem from the fact that the 26-year-old feels well settled in Manchester and is enjoying his time at the club.
The former Atletico goalkeeper, who was named United's player of the season three times between 2013 and 2016, has not conceded a goal at Old Trafford since April.
He kept his 100th clean sheet for United in the 4-0 win over Everton on 17 September.
De Gea said in an interview earlier this month that he was keen to prove himself as the best goalkeeper in world football.
"I always want to be the best," the Spaniard told United's official website. "I train hard to improve every day, in every training [session], so I work really hard to be the best keeper in the world. I don't know who is the best at the moment but this is what I want – to be the best.
"All three keepers at United – Sergio [Romero], Joel [Pereira] and me – train really hard to be the player who is on the pitch to play in the games.
"Now I feel really strong, really confident. Everyone trains hard to be in the XI and that's it."