The former Manchester United winger suffered the injury to his left knee in the 29th minute after clashing with Hamari Traore but was only replaced at halftime by Martin Terrier.
The Groupama Stadium outfit also lost Jeff Reine-Adelaide to a similar injury in the first half and he is also likely to miss the rest of the campaign.
Lyon went on to lose the game 1-0 which leaves them in eighth place on the league log, five points adrift the European places.
Depay is expected to miss the remainder of the season, putting his involvement at EURO 2020 in jeopardy.
"It's a very bad evening because we lost and because of the lack of quality. The very bad news is also the injury of Memphis, who could be out for a while," Lyon sporting director Juninho said.
"He's our captain and I'm sad for him."
The 25-year-old has arguably been Lyon’s best player this season, scoring nine goals and providing two assists in 13 Ligue 1 appearances so far.
Depay has also been instrumental in the Champions League and has scored five goals in the competition, including the equaliser against RB Leipzig last week that sealed Lyon’s place in the knockout round.
Depay was instrumental for Netherlands during the EURO 2020 qualifiers but Ronald Koeman might have to do without him in a group that includes Austria, Ukraine and a third member to be confirmed after the playoffs early next year.
Lyon will find out who their Champions League round of 16 opponents are in Monday’s draw in Switzerland.
