Solskjaer has been a revelation since replacing the sacked Jose Mourinho on a caretaker basis in December.
He has led them to 14 wins from 19 games in all competitions.
Manchester United have reached the quarter finals of the Champions League and now have a realistic chance of securing a top four finish in the Premier League.
Despite the success, United bosses remain determined to wait until June before deciding who the new permanent manager will be.
But Pogba is desperate to see Solskjaer get the role and believes he has done more than enough to deserve it ahead of the likes of Mauricio Pochettino and Massimo Allegri.
Pogba said: “Obviously the results have been very good and some of us have a coach that we know already from before.
“We want him to stay. The results have been great and he has a great relationship with the players.
“Of course I want him to stay. I have a great relationship with him, so yeah, he totally deserves it. He knows the club. We want him to stay.”
“We want him to stay. The results have been great and he has a great relationship with the players”
Man Utd star Paul Pogba on Ole Gunnar Solskjaer
Pogba endured a difficult relationship with Mourinho and has taken a swipe at the Portuguese by claiming Solskjaer has made playing football feel like fun again.
The French World Cup winner added: “He’s a really, really happy coach that has given the confidence back to the players.
“He has given us the freedom to play and enjoy our football again. Maybe we lost that with the results we had before.
“He has just told us to enjoy playing football again, because when you don’t, there is no point playing. So enjoy doing what you love.
“When a player is happy he wants to keep being happy – and he has a great relationship with all the players.”