United bosses have held talks about a new contract with De Gea, whose current deal runs out at the end of next season.
Keeping the Spanish international, 28, is crucial to United’s chances of future success.
He has been a long-time target for some of Europe’s biggest clubs, including Paris Saint Germain, Juventus and Barcelona.
Starsport understands De Gea is hoping to land a new deal worth around £350,000-a-week – making him the highest paid keeper in the world.
Caretaker boss Solskjaer reckons De Gea is up there with the all-time greats – and would hate to see him leave Old Trafford.
He said: “There are no updates from me, but the club and David are in dialogue.
“Hopefully they’ll agree, it’s out of my hands but I’m sure we’ll do what we can.
“David’s done fantastic since he came, the club really showed how much they wanted him back in the day.
“There was probably keepers out there better at that time, but the belief the manager showed in him with the criticism he got, has now paid dividends.
“He’s been ‘Player Of The Year’ I don’t know how many years and we’re grateful to have him.”
Solskjaer reckons De Gea is even better than United legends Peter Schmeichel and Edwin van der Sar and he added: “It’s difficult to compare them, but Edwin and Peter were never player of the year.
“David’s won so many points for us over the years and I would say he’s the best goalkeeper in the world.
“Peter was at the time and Edwin at the time as well. I’m delighted with David and his personality is fantastic. He’s such a humble and confident boy.”