Real president Florentino Perez and under-fire boss Zinedine Zidane spent all summer trying to prise the World Cup winner away from Old Trafford.
But Solskjaer stresses: “I am not worried now and I will never be worried in the future about Paul Pogba staying here.
“If it all kicks off again 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."
Solskjaer continued :“I watched their game at Paris St Germain and it was a big defeat.
“Meanwhile 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."
Solskjaer has spent months showing Pogba how much he likes and respects him.
He says: “You can ask Paul if our relationship is helping him want to stay here.
“I certainly hope so because he is a great player and we want to see him at his best in a United shirt."