Real tried to get Pogba, the France midfielder, in the summer but their offer – £27.6m plus Colombia midfielder James Rodriguez – was deemed well below United’s valuation of the World Cup winner.
Zinedine Zidane, the Real head coach, is reportedly still set on getting Pogba, especially after they were outclassed in Wednesday’s 3-0 Champions League defeat at Paris Saint-Germain.
United may struggle to convince Pogba, who said it “could be time for a new challenge” in the summer, to stick around at Old Trafford, but Solskjaer insists he is here to stay.
“I don’t listen to the president of Real Madrid [Florentino Perez],” said Solskjaer ahead of Sunday’s trip to West Ham, live on Sky Sports.
“But Paul is working hard here and he is totally committed to Manchester United.
“We have seen that while he’s been injured. He has been working like crazy to get himself fit because he wants to play for us and he wants to help his team-mates.
“I am not worried now and I will never be worried in the future about Paul Pogba staying here.
“If it all kicks off against with Madrid in January then there will be no worries then either. If we have to live with the speculation again then we will handle it like we did in the summer.
“Paul is going nowhere. He is happy here when he’s playing.”
Despite Pogba’s flirtation with Real – he said it would be “a dream for anyone” to play under Zidane at Real – Solskjaer remains fond of the player he coached as a youngster in the United reserves.
“You need a big personality to play for Man United,” said Solskjaer. “You need a certain kind of confidence and arrogance and you have to be self-assured enough to be yourself in both the good times and the bad times.
“Paul Pogba is all of those things. He is one of our big characters. Paul will walk out onto any pitch and be full of confidence – and that’s what I look for and what I demand from all of our players.
“It’s not misplaced arrogance with Paul. It’s about believing in himself. It’s about the respect he has for his team-mates and his opponents.”