Greenwood, 18, has scored 15 goals in 41 appearances in his first full season for Manchester United and broken into the starting XI since football resumed last month.
Bradford-born goalscorer Greenwood only made his England Under-21 debut in September and has attained four caps ahead of the U21 Euros in Hungary next summer.
Southgate has a myriad of options to choose from in attack but Greenwood could enjoy playing time in the Champions League next season just before the Euros start next June.
“When you play for Man United, I think that’s a test that’s good enough and difficult enough,” Solskjaer said. “If you can handle playing for Man United then you can handle playing for any team and Mason’s got the qualities that not many others have.
“So he could play for the England full team or Under-21s, I don’t really mind. I know he’s doing well for us and it’s great to have him here. Mason’s a naturally good finisher, so he’ll always score goals.
“He’s a humble boy, he wants to learn, he listens, he’s worked hard; you can see his fitness is getting better and better. He’s just a young boy who will fill out. He’s exciting to work with and we’ll keep on building him up in the gym and on the pitch.”
United are interested in signing England international Jadon Sancho, who would likely start in the right-sided attacking role Greenwood has occupied in the last three league games.
Borussia Dortmund winger Sancho, 20, was expected to start on the right of England’s attacking trident with Harry Kane and Raheem Sterling had the Euros taken place this year.
England’s next fixture is in the Nations League against Iceland on September 5.