The striker, 17, scored in a 4-0 pre-season friendly over Leeds in Perth.
United’s most experienced striker Romelu Lukaku, a £75m signing in 2017, is a reported Inter Milan target.
“Mason has a chance to start against Chelsea [on Sunday, 11 August], definitely. He is more than capable,” said Solskjaer.
“It is a good possibility if he keeps going as he does. It is difficult to keep players out who perform well.
“[I was] delighted with the goal he scored. That first goal is a weight off his shoulders. With his movement, cleverness, he’s going to be a very good player. He’s a natural footballer who can play anywhere along the front line.”
England Under-18 forward Greenwood made his United debut in March’s Champions League comeback win over Paris St-Germain and has made three Premier League appearances.