Martial and Lingard both limped off during the 2-0 defeat to Paris Saint-Germain in the Champions League on Tuesday and are set to miss the FA Cup fifth round tie with Chelsea at Stamford Bridge on Monday and the Premier League clash with Liverpool at Old Trafford six days later.
"I think they'll be out for two to three weeks," Solskjaer told a news conference on Friday. "Then you're looking at young players and it's Mason Greenwood's time to step in but he's injured unfortunately, out for a couple of weeks as well.
"Sometimes that's how the luck goes but Mason will get his chance."
Martial's groin injury and Lingard's hamstring problem will rule them out of games against Chelsea and Liverpool, as well as the trip to Crystal Palace on Feb. 27 and a home game against Southampton on Mar. 2.
The pair will also be doubts for the return leg against PSG in Paris on Mar. 6 and the trip to Arsenal on Mar. 10 but Solskjaer, who has won 10 of his 12 games in charge, said their absence will present opportunities for Romelu Lukaku and Alexis Sanchez.
"It's a chance for anyone who plays," Solskjaer said. "Alexis, Romelu, Chongy [Tahith Chong] or Angel [Gomes].
"Anthony and Jesse have done really well for us, they're vital but then again with Rom and Alexis, they've got different skill sets and will be able to show what they can do, so we need to gel a team together for Monday."
Solskjaer oversaw defeat as United boss for the first time against PSG on Tuesday. He returned to Norway to spend time with his family after the game before heading back to Carrington to prepare for games against Chelsea and Liverpool.
With injury problems and two tough games, it will be a testing week for Solskjaer but the 45-year-old has a different view.
"It's a great week," Solskjaer said. "You don't panic because you've lost a game. You don't change everything, it doesn't change the mindset at all.
"We need to keep working the way we play football, we want to go down to Chelsea, we want to have Liverpool here, and impose ourselves on those games.
"There are quite a few other teams with injuries so you have to adapt to that."