The Red Devils have seen a resurgence in form since the arrival of Solskjaer, who replaced Jose Mourinho on an interim basis.
They have won all seven games that he has taken charge of – including a 1-0 victory over Tottenham.
But they face a whole new challenge tomorrow night when they take on Arsenal at the Emirates in the FA Cup.
Away from the pitch, United are not expected to make any transfers this month – instead planning for the summer.
And according to the Metro, Solskjaer has provided an analysis on the current squad in the latest hint he could land the full-time job.
A number of managers had been linked with the Old Trafford position.
Some include Mauricio Pochettino, Zinedine Zidane and Laurent Blanc.
But Solskjaer has forced himself into serious consideration as a result of his philosophy on the pitch as well as his ability to install a happy camp.
The Metro report that Tottenham boss Pochettino remains the top target, but with a huge summer expected some deals will need to be discussed immediately.
They say the United board requested his opinions on the United squad he took over from Mourinho and where they could now strengthen.
He said: “I’m happy with squad I’ve got.
“I don’t expect anyone to come in but obviously if the club have targeted and identified a few and I think they might fit here, I’ll give my thumbs up. But it’s not just my decision.”
Meanwhile, United are keen on pushing ahead with contract talks with the likes of David de Gea, Anthony Martial and Juan Mata.
Solskjaer revealed he hoped that there was good news in the coming weeks and that talks are being held.
“I think the club’s making progress,” he said.
“But I leave that to the right people to do that.
“I don’t know really how far we are away from it, but hopefully we’re going to have some good news in the coming weeks.”