The French midfielder has been linked with moves away from the club in recent years, but following his recent return to action and impressive form, it's been suggested that Pogba could agree a new deal at Old Trafford.
Asked about the prospect of Pogba committing himself to United for the long term, Solskjaer replied: "Of course we want to keep the best players around the club and hopefully we can do something.
"We want to build a squad for the future. We have secured one at the start and another who is really experienced. But we also need players in the mid-range group.
"Since Paul has come back he's looking better and better. He's enjoying his football so let's see where it takes us."
United have been linked with a raft of players in recent weeks, including the likes of Jadon Sancho and Jack Grealish.
But Solskjaer has tried to temper fans' expectations, admitting that the coronavirus pandemic has changed football's economic landscape.
The United boss - who replaced Jose Mourinho in the role in late 2018 - explained: "The whole world has changed both financially and the perception we've got on values.
"Every time I put the case in front of Ed [Woodward [the club's executive vice-chairman], I think it's a sensible one and a realistic one. I've always got the club in the front of my mind.
"I don't think short term, I always think long term and try to make good deals. I'm always quite careful with money."