Solskjaer is in temporary charge until the end of the season having replaced Jose Mourinho at Old Trafford.
He has got off to a flying start as Manchester United manager, winning eight of his first nine matches in all competitions.
Solskjaer is yet to manage in the Champions League for United, with the Red Devils facing Paris Saint-Germain in the last 16.
United face the Ligue 1 giants across two legs with the first of those at Old Trafford on February 12.
And The Sun report that Solskjaer will be handed the permanent United job if he gets past PSG.
Red Devils executive vice-chairman Ed Woodward reportedly held talks with Ole Gunnar Solskjaer last week.
And though it’s claimed there were no assurances of a permanent offer yet, it’s believed that more positive results will only enhance Solskjaer’s credentials.
The Norwegian has done enough to change United hero Paul Parker’s opinion on whether he should be the next permanent boss.
“Initially on this show I turned round and said about Ole that I worry about him attracting players,” Parker told talkSPORT.
“He’s a legend with Manchester United but I was worried about him attracting players to Manchester United.
“But what he’s gone and done says to me he has definitely jumped up high [in the running for the job].
“I believe he could manage Manchester United next season and beyond.
“But I think he’s got to have a director of football, a sporting director with him.
“He’s already got the godfather [Sir Alex Ferguson] behind him already. The boss is involved, he’s there, definitely there.”
“And maybe a sporting director, somebody who works with all these agents who knows this man and knows the line he’s going down.
“I think he can do that job if he’s got the right people around him.
“We’ve seen it already, he’s got Mike Phelan next to him, people see him sitting there looking quite relaxed but he’s got a major say on what’s happening on that field of play.”