The new Red Devils boss has had a stunning start to his reign at Old Trafford, winning eight and drawing one of his nine matches in charge.
Solskjaer has turned the mood around at the club, steering them back into the top four mix after Jose Mourinho's disastrous run.
The Norwegian suffered his first blip in charge on Tuesday night, following a 2-2 home draw against Premier League strugglers Burnley.
But even that impressed members of the board, coming back with two late goals to salvage a draw after being 0-2 down with five minutes to go.
It is understood United chief Ed Woodward held talks with the new boss last week at the club's Carrington training complex.
He is thought to have passed on his congratulations for the excellent job so far, and offered up his services if anything was needed.
And the majority of the Red Devils squad and members of the backroom staff believe Solskjaer will land the job on a full-time basis if they get past PSG in February.
An Old Trafford source said: "Woodward spent around half an hour with Ole at Carrington in the middle of last week.
"He was passing on his thanks from the board and seeing if he needed anything.
"There was a little appraisal of the squad and mood.
"There was no assurances of a permanent offer as of yet.
"Woodward is fully in Ole's corner but the Glazers have the final say.
"He pushed for Jose, so has some making up to do. The consensus around the training ground is Ole will get the job though with a positive result against PSG."
Solskjaer has turned United's fortunes around so much, five first-team stars are now set to sign new long-term contracts.
Anthony Martial has already pun pet to paper while David De Gea, Paul Pogba, Marcus Rashford and Ander Herrera are aset to follow suit.
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