As a result, they will miss the club's FA Cup fifth-round clash with Chelsea on Monday as well as a Premier League battle with title contenders Liverpool on February 24.
One possible replacement named was striker Mason Greenwood but the 17-year-old is also injured, preventing him from making his first-team debut in their stead.
"I think they'll be out for two-to-three weeks," Solskjaer said of Lingard and Martial.
"Then, you're looking at young players, Mason Greenwood, it's your time to step in, and 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."
Lingard was forced out of the 2-0 defeat to PSG late in the first half and was replaced by a relatively ineffective Alexis Sanchez.
Solskjaer himself said there was little he could do to help the struggling Chilean, who failed to make an impact in his appearance.
Martial, meanwhile, was removed at half-time for Juan Mata after carrying an injury through the first half, as Solskjaer was forced into making two changes.
Sanchez and Romelu Lukaku will likely be given chances to prove themselves in the coming weeks while Solskjaer also says Tahith Chong and Angel Gomes are in contention.
"Chance for anyone who plays, Alexis, Romelu, Chongy or Angel," Solskjaer said.
"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 just need to gel a team together for Monday."
Then comes the Champions League return leg in Paris on March 6 when the Red Devils look to overcome their two-goal deficit without the likes of Paul Pogba, who will miss out after being sent off in Tuesday's defeat at Old Trafford.