Pogba said a "question mark remains" over his future following United's 4-0 rout of Chelsea in their Premier League opener on August 11.
While the Premier League's transfer window has already closed, the LaLiga and Serie A markets remain open until September 2 – prompting ongoing speculation about a possible move to Madrid or a sensational return to Juventus.
Solskjaer, however, has no concerns over star midfielder Pogba ahead of United's trip to Wolves on Monday.
"You [the media] always put question marks around Paul," Solskjaer said. "I don't think it's odd saying enjoy playing, I have fun with my team-mates, I enjoy what I'm doing, I love my job and enjoy the game.
"Of course that sentence with the question mark about him, there's always question marks about Paul.
"There's not one press conference that I've not answered a question about Paul Pogba – 80 per cent of what he said was that he enjoyed that game and his time here.
"I've got absolutely no concerns on Paul. For me, yeah, he's going to stay."
Meanwhile, Solskjaer admitted United have "too many" centre-backs following the world-record arrival of Harry Maguire from Leicester City.
With Maguire now at the Theatre of Dreams, Solskjaer's options also include Victor Lindelof, Axel Tuanzebe, Chris Smalling, Phil Jones, Marcos Rojo and Eric Bailly, who is recovering from a serious knee injury.
Argentina international Rojo was linked with a deadline-day move to Everton but a transfer did not materialise.
"I do have too many centre-backs to keep everyone happy," said Solskjaer. "But we need to win games, we need to perform, so we're going to pick the players that will give us the biggest chance to win games."
"They're all quality centre-backs and I know that we will need quite a few of them," added Solskjaer. "Of course, Eric's injury means that at the moment I've got six fit ones, which is maybe one too many to keep everyone involved.
"But then that's our jobs, to make the most of what we have now, and for them as well to make sure every day in training.
"And when they get the chance, because they will – we've got loads of games, play well when you get it. All of these, Marcos, Phil, Chris – they've been here for a long time and they've proven that they can do it."