Solskjaer has won 10 out of 12 games since replacing Jose Mourinho on an interim basis in mid-December – the only blips being the Premier League draw against Burnley and Champions League defeat to PSG.
The Norwegian’s loan contract from Molde expires at the end of the Premier League season in May.
Molde have insisted Solskjaer will return to his role at the club but Manchester United’s fine form has thrown a spanner in the works.
He has emerged as the leading rival to Tottenham head coach Mauricio Pochettino to land the gig on a full-time basis.
And, according to The Sun, Solskjaer has already been guaranteed the job by executive vice-chairman Woodward.
It is claimed the pair enjoyed a three-hour dinner at Italian restaurant Cibo in Hale, Cheshire on February 6.
The Sun report that during the meal, Woodward shook Solskjaer by the hand before telling him: “Here’s to a long and successful relationship together.”
They reportedly left in club cars after Woodward settled the bill.
A witness told The Sun: “When me and my party got to the restaurant Ole and Ed were already there.
“They spent the whole time eating but mainly talking quietly.
“There were big smiles between them as they got up to go and shook hands before getting into separate Mercedes that were parked on the kerb outside.”
Solskjaer’s former United team-mate Paul Scholes has backed his pal to get the job.
“I’d have no issues whatsoever,” he said.
“With what he’s done for these last 11 games – I know people will start to question because now because he’s lost a game but that was PSG.
“That could happen to the most experienced manager in the world because PSG are a top team.
“I think for what he’s done so far he definitely deserves a chance.”