Executive vice-chairman Ed Woodward has previously said a decision would not be made until the end of the season.
However, The Sun claims he is ready to change his mind and bring that forward given the Norwegian’s impressive record since he took over from Jose Mourinho as interim boss in December.
Despite suffering back-to-back defeats against Arsenal and Wolves, Solskjaer has led United to an impressive 14 wins in his 19 games in charge.
He is also credited with a dramatic upturn in from of United’s key players such as Paul Pogbaand Marcus Rashford who were visibly struggling under Mourinho.
It is hoped the appointment of Solskjaer will hand the club a boost in their crucial run-in between now and the end of the season.
They are currently fighting it out with Arsenal, Chelsea and Tottenham for a place in the top four in the Premier League.
United also have the small matter of a Champions League quarter-final with Barcelona next month.
The Red Devils players have been unanimous in backing Solskjaer to be given the job on a full-time basis.
“Of course we want him to stay,” midfielder Pogba said last month.
“The results have been great. I have a great relationship with him, he has a great relationship with the players.
“When a player is happy, he wants to keep being happy. Solskjaer deserves it. He knows the club, he knows everything about the club.
“He is a really happy coach that gave confidence back to the players.
“This gave us the freedom to play and enjoy football again because maybe we lost that with the results that we had before.”
And Rashford was full of praise for his boss when quizzed last month.
“For me, it’s been a massive help having that type of guidance around me,” he said.
“But he’s been working with all of the players with everything he’s got, you can see even the defenders are improving, the midfield is improving, the team’s improved overall but we understand we can still take more steps forward.
“He has that know-how, the understanding, that’s the kind of thing you just pick off naturally off these type of people.
“Even not just out on the pitch, having conversations about it, it all adds up.
“It’s been a big turnaround. As long as we keep improving I don’t see why [he shouldn’t get the job permanently].”