Pogba's agent Mino Raiola has said publicly the midfielder wants to leave United, either next summer or at the end of his contract in 2022.
"There's always a demand at Man United to win trophies and the more you win trophies, the more players are going to want to play for us and then there's a higher standard you've got to set yourself," Solskjaer told ESPN in an exclusive interview.
"It's a demand for the players, it's a challenge for the players that we want them to improve and get better and better. Paul has been here for a while and of course he wants to win more trophies.
"It's down to us to win trophies and then let's see who will be part of the team going forward."
Raiola's comments have sparked a swell of speculation about Pogba's next move, although United have not yet given up hope of agreeing a new deal with their record signing.
Pogba is set to start against Everton in the Carabao Cup quarterfinals on Wednesday and Solskjaer insists the World Cup winner has not been distracted by talk surrounding his future.
"He's got a very, very good attitude to the training and his football," Solskjaer added.
"He loves football and he's ambitious and wants to have success like most of the others -- most of the others that are in the squad. We're not all the same but I would say that this squad has impressed me with their attitude, spirit and the way they go about their work."