Greenwood has enjoyed a superb breakthrough season, scoring 14 goals in 41 appearances across all competitions.
He scored one and provided another in Tuesday’s impressive 3-0 win at Brighton and Solskjaer is excited to see what else he can do.
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“We have done well over the course of the season giving him more and more exposure, more and more game time,” the United boss said.
“You know, it’s not easy being a young boy. He could still play in this season’s FA Youth Cup team, so he’s a young boy that we have to look after. But he’s performing from the bench or when he starts and he’s developed fantastically this season.
“Only the sky’s the limit. It’s up to him. He knows what he should do, can do and must do to keep on playing, but that’ll be between us.”
United were interested in adding another talented English teenager to their ranks in the form of Jude Bellingham, but the Birmingham midfielder appears to be on the verge of joining Borussia Dortmund.
Asked if it was frustrating to miss out on the 17-year-old, Solskjaer said: “Well, I think there’s been loads of speculation and linked us to every single talented player from the age of 15 to 25, really.
“I can’t comment on every single player that’s been linked with us so, no, I can’t really comment.”
Meanwhile, youngster Ethan Laird has been called into the training squad with fellow defenders Phil Jones and Axel Tuanzebe sidelined.
“Phil and Axel will still be out for a little while,” Solskjaer added.
“It’ll be a few weeks before Phil can start training and Axel, we won’t probably see until September. Ethan is going to join the training group now.”