It emerged in July De Gea, 28, had verbally agreed a six-year contract with United worth £375,000-a-week and the club expected him to sign new terms upon their return to Europe from the pre-season tour of Australia and Asia.
However, De Gea is still yet to commit and about to enter the last nine months of his existing United deal. If De Gea has still not inked a new contract before the January transfer window he will be free to negotiate a pre-contract agreement with overseas clubs.
Paris Saint-Germain approached De Gea's camp in the summer but eventually signed Keylor Navas, 32, from Real Madrid on deadline day. PSG could return for De Gea in 2020 and Juventus have also been linked with the Spain international.
Solskjaer has installed De Gea as the first-team captain this term and remains hopeful the Spaniard will extend his stay at United.
“I think you'd need to ask David and the club that," Solskjaer said about the De Gea delay. "I've obviously not been involved in all the discussions, but I'm very - or pretty - confident we'll get this sorted.
"David's never, ever said anything else to me, other that he loves this club, he wants to stay here and he's had a fantastic time, so let's hope we just get it over the line.
"I think he's had a fantastic time here, but yeah, of course, whenever he concedes a goal, it's headlines, that he should have saved. Whenever we lose a game - we've lost one game since July 1 - and it sounds like we're really struggling.
"But that's just the way it is at Man United and you have to handle it. You have to be resilient, you've got to have robust confidence, and he's fine. I don't worry about him at all."
Atletico Madrid 'keeper Jan Oblak has shown interest in joining United in recent years and the club tied down academy goalkeeper Dean Henderson - on loan at Sheffield United again - on a three-year contract in July. United's number two goalkeeper Sergio Romero has often impressed whenever he has replaced De Gea and the 36-year-old Lee Grant is still the club's third-choice goalkeeper.
Romero is contracted to United until 2021 and the Henderson renewal was additional proof the club has taken measures to prepare in the event De Gea does depart next year.
"Well we are always aware of what’s happening around in the world," Solskjaer added. "But my focus and the club’s focus is to convince him to stay. [De Gea] can see what we’re doing, and knows what we’re doing.
"We want players here, the best ones, and the ones we want most of the time stay. There are not many who we want to keep that we don’t manage to keep.
"I have two very good backup keepers with Sergio and Lee. Dean is playing at Sheffield United. So the goalkeeping department is in good health.
"And I said I am confident we will manage to convince David that here is the future for him. And it is a successful future, a club that wants to get back to winning trophy."
When asked whether retaining De Gea would be a statement their best players do not want to leave, Solskjaer replied: "Yes, definitely.
"We have gotten used to the best keepers here, [Peter] Schmeichel, De Gea, we want the best ones. [Edwin] van Der Sar is getting too old now, he did okay in the [Vincent Kompany testimonial] game, and now for me we want to build a winning team again, to challenge for trophies."