Greenwood, 18, made his senior debut against Iceland on 5 September having made 50 appearances in all competitions for United last season.
"I tried my best to give him a rest over the summer," said Solskjaer.
"The club, we asked specifically and pointed out that he should have had a rest.
"He needs it mentally and physically after that season he had. First thing that happens is, one, he's called up, and two, he's right there in the press.
"We've done whatever we can to protect him and I will keep on helping him."
Forward Greenwood apologised for the "embarrassment he had caused" in Iceland, and later had to release a statement about his "poor judgement" when he inhaled nitrous oxide, after "historical pictures" of him doing so were published in a newspaper.
Solskjaer confirmed that Greenwood will be involved in United's Premier League season-opener against Crystal Palace on Saturday.