The Spain international has been offered a new deal worth around £350,000 per week.
It’s hoped he will put pen to paper after United’s pre-season tour.
Once the 28-year-old is signed up, United will then allow Dean Henderson to pack his bags.
The Times claim Ole Gunnar Solskjaer is happy to sanction his departure, with Sheffield United the most likely suitors.
READ MORE: Man Utd agree £80m Harry Maguire transfer
“Ole Gunnar Solskjaer wants David de Gea to pen a new deal”
Henderson, 22, spent last season on loan at Bramall Lane and impressed.
He racked up 46 appearances as the Blades booked their place back in the top tier.
Solskjaer also has Sergio Romero on his books.
The Argentina international is an experienced campaigner and has been De Gea’s understudy for the last four seasons.
United, meanwhile, continue to be linked with Leicester defender Harry Maguire.
However, club legend Denis Irwin says a deal is far from done.
“No, no, Harry is a Leicester player isn’t he,” said the former left back when asked if Maguire had signed.
“Listen, we finished really, really poorly last year. When Ole took over, we were really struggling.
“We were 11 points behind fourth place, and we got ourselves up there, got ourselves to the quarter-final of the Champions League, and we fell away badly in the last two months, six weeks of the season.
“This is Ole’s first pre-season, so he has got a chance to stamp his identity on the football team and the way he wants Manchester United to play.
“We’ve brought in a couple of young lads, who have got plenty of pace – I think something that we lacked in the last two-three years – and I think we need another couple of signings.
“It would be great to get another few signings in and go from there because it’s a fresh start, a clean slate.”