Anthony Martial (thigh) and Paul Pogba (ankle) have been ruled out by Solskjaer, who sold striker Romelu Lukaku to Inter Milan and sanctioned winger Alexis Sanchez's loan to the Italian side. Jesse Lingard (illness) is also a doubt for Leicester's visit.
Greenwood, 17, impressed on his full debut in the final game of last season against Cardiff and came on in all four of United's Premier League fixtures last month. The Bradford-born forward has had just 47 minutes of playing time this season but is in contention to replace Martial.
Nineteen-year-olds Tahith Chong, named on the bench at Southampton two weeks ago, and Angel Gomes are other attacking options to make the matchday squad since United could be without seven players overall.
"I'm prepared to start any of the players in this squad," Solskjaer said when asked about Greenwood. "They are in the running and we just have to make the last call after training today and tomorrow morning.
"We have a squad of 24, 25 players, we expect all of them to play their part at certain times, different times in the season. This weekend might be one of them."
United inflicted a 4-0 drubbing on Chelsea in their Premier League opener last month but have managed just three goals since and struggled to fashion clear-cut chances.
With Martial unavailable against Leicester and Lingard a doubt, the United attack could consist of a 17-year-old, a winger with three top-tier starts to his name, veteran playmaker Juan Mata and Marcus Rashford, who does not regard himself as a number nine.
"You have 38 games, so if you can have 19 goals [each] as a partnership, with those two, so around about 40 goals, that's okay," Solskjaer added. "But it's not top, top class.
"At the moment, Dan [James] has got three, Marcus and Anthony have got two each, and that's in our four games, and Anthony only played three of them, so he's improving.
"It's anticipation, it's relationships with the players, knowing your players, knowing which position you're in. Because the quality is there, definitely and we're learning where to run, when to run, how to run, and the last finish, they've go to do, of course.
"And they're practising all the time and hopefully we can see Dan and Marcus getting goals this week, and Anthony can come next week."