Manchester United are getting themselves ready for Saturday's clash against Southampton following Wednesday's win over Crystal Palace.
Vice-chairman Woodward was one of the first to arrive at Carrington after he reportedly gave Solskjaer the go-ahead to take charge permanently.
The pair shook hands over the deal during a three-hour dinner at a swanky Italian restaurant this month.
United are not expected to make an official announcement until the summer, but Woodward still has work to do behind the scenes.
Solskjaer has already moved out the Lowry Hotel as sources claimed he is looking for somewhere more permanent.
A number of top names were tipped to take over in the summer, but the 46-year-old's run of results have made him the obvious choice.
The Reds are unbeaten in the Premier League since the Norwegian took over from Jose Mourinho in December.
Woodward will also be hoping to resolve the contract dilemma surrounding De Gea after Solskjaer admitted an agreement has not been reached.
United's No1 goalkeeper has been locked in talks over a new £350,000-a-week long-term contract.
But Solskjaer confirmed at the pre-match press conference that United are trying to thrash out terms with De Gea.
He said: "No updates from me, the club and David are in dialogue, hopefully they'll agree, it's out of my hands but I'm sure we'll do what we can.
"David's done fantastic since he came, the club really showed how much they wanted him back in the day when Eric and the gaffer found him.
"There was probably keepers there better at that time, but the belief the manager showed in him with the criticism he got paid dividends.
"He's been the player of the year for I don't know how many years, we're grateful to have him.
"Then again we've got Sergio [Romero] and Lee [Grant], fantastic goalkeeper depth, I don't think there's any manager in the world who'll beat me in that respect."
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