But the future of captain Antonio Valencia is less clear with the club apparently poised not to trigger a contract extension for the long-serving Ecuadorian.
United did take up an option to extend De Gea’s current deal to 2020 this season.
However, reports suggest the Spaniard is holding out for an offer at least on a par with Alexis Sanchez as United’s highest paid player having won the club’s player of the year for four of the past five seasons.
Solskjaer is optimistic that United will take care of the business before running the risk of losing De Gea for a cut-price at the end of the season or on a free transfer next summer.
“There are no updates from me, 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,” said Solskjaer.
“He’s been player of the year I don’t know how many years, four or five, we’re grateful to have him.”
Solskjaer would not be drawn on ranking De Gea alongside his former teammates Edwin van der Sar or Peter Schmeichel from his playing days but clearly feels that United’s number one is worthy of comparison with the club’s best ever.
“It’s difficult to compare them but Edwin and Peter were never player of the year,” he said.
“David’s meant 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. Such a humble and confident boy.”
The future looks less rosy for Valencia, currently struggling with a debilitating calf injury, the latest in a long list of fitness problems that have limited him to one league start in the past five months.
The 33-year-old full-back, who will have been a United player for 10 years this June, is out of contract this summer and it appears that the club will not activate the one-year option that could keep him at Old Trafford for an additional 12 months.
That deadline was due to expire at 5pm on Friday, but Solskjaer has not ruled out Valencia remaining at Old Trafford if he can prove his fitness over the final weeks of the season.
“He’s still not 100 percent fit, he is still working to get back fit and he’s had a fantastic career here at Man United and in England,” said the manager.
“He is one of the Premier League winners left in the dressing room, but at the moment I’m not sure if Man United and Antonio will agree on the next year. It depends if he gets back on the pitch as well in the next few months.
“But he’s the captain, a great servant to the club so hopefully I can get him on the pitch and show what he can do, or he can get himself 100 percent fit and back on the pitch.”
Solskjaer is aiming to stretch his unbeaten run in domestic games to 15 matches when Southampton visit Old Trafford on Saturday.
The Norwegian is still without a host of first team regulars through injury, but Marcus Rashford has shaken off the effects of an ankle injury and is set to start.