The Norwegian has revitalised United since taking over in December, winning 12 of his 15 games in charge at Old Trafford.
The interim boss has not yet been informed whether or not he will be kept on beyond the end of the season – though his involvement in advertisements for 2019-20 season tickets at the club suggests he could well still be around come August.
Yet Solskjaer has revealed that he has had private discussions with key United chief Woodward over which players should stay or not for next season, regardless of his own personal future.
That’s with the club have decided against exercising the one-year extension clause in captain Antonio Valencia’s contract this week, freeing the defender up to leave United in the summer.
The Red Devils are otherwise working on new deals for David de Gea, Marcus Rashford, Ander Herrera and Juan Mata and have already extended the stays of Anthony Martial, Ashley Young and Phil Jones since Solskjaer came in.
Asked if he had discussed the state of his squad to Woodward, the caretaker boss told reporters: ”Yes, I have.
“We’ve been discussing the future of the club – we did that before the talk of, ‘If I should stay or not’.
“My contract is up in the summer but as I’ve always done for United I give my best because you have to plan, even if I’m not here I still give my view on who I would like to see here as a supporter.”
And club legend Solskjaer – who has proved an immensely popular appointment among United fans – reiterated his desire to be handed the job long term.
He added: ”I’d like to stay, of course, but that is not decided until the end of the season.”
United were 11 points off the Champions League spots when Solskjaer arrived at the club but have cut the gap to the top four to a single point going into today’s visit of Southampton (3pm).
And the 46-year-old thinks it is essential United are in Europe’s top-tier competition next season to attract more top-level stars to rival their current options.
He added: “We are hoping and planning that we will get into the Champions League for next year. We’ll take that as it comes.
“All the players want to play in the Champions League. I think to play in the Champions League for Manchester United is a bigger thing for many and we know we can’t live off our name and tradition and history.
“We want to attract good players that want to improve the players we have here: Paul Pogba, David de Gea, Anthony Martial and Marcus Rashford.
“All these top players we have – they need players to challenge them because no one should feel 100 per-cent secure to be in the starting XI.
“We are five points behind Tottenham and there’s 10 games to go so definitely we’ve got a chance to move up the table so we will be doing our best.”