United have been in fine form since sacking Jose Mourinho a week before Christmas.
Much of that has been down to Solskjaer’s management, which has seen the club turn their fortunes around.
When the Norwegian took over, the Red Devils were 11 points off a spot in the top four of the Premier League.
But prior to last weekend, United had found themselves back in a Champions League spot – but then Arsenal came along and beat them.
Before that defeat to the Gunners, the Old Trafford outfit had gone unbeaten in the league under Solskjaer – including a new away record.
They have also claimed the scalps of Arsenal and Chelsea in the FA Cup and managed an incredible result in the Champions League.
They were 2-0 down to Paris St-Germain in the first leg but clawed back an unlikely 3-1 result in the Parc des Princes to make it into the next round.
And Solskjaer’s No.2 Carrick believes the club should remove any doubt and hire him on a permanent basis as soon as possible.
“Of course I want him to stay, for me to say anything else would be ridiculous,” Carrick said.
“It’s obvious that all the players and everyone wants Ole to stay, especially when you’re winning games as well the players always want the manager to stay.
“That’s the factor of any United fan or staff, I think if you asked anyone really no one is going to go against Ole at the moment.”
Meanwhile, the Red Devils face Wolves next in the FA Cup at the Molineux tomorrow.
United have been struck by a number of injuries but Solskjaer has been positive about their returns.
“[Lukaku] had a sore foot this week he’s being assessed, so hopefully he’ll get through today,” Solskjaer added.
“Phil Jones is back, there have been so many players out, Anthony (Martial) is back, Ander (Herrera) has been training, Jesse (Lingard) has been training and we’ve got a decision to make tomorrow on who’s match fit.”