The 26-year-old was injured when a car he was travelling in left the A28 motorway around 60 kilometres north-east of Rouen. L'Equipe's report claims the driver of the vehicle had fallen asleep at the wheel, although that has not been confirmed.
Anin reportedly suffered arm and a spinal injuries and was placed in an induced coma at Rouen University Hospital. Three other people were also taken to hospital following the accident, which occurred in the early hours of Tuesday morning.
"In the light of the information currently in our possession, the situation seems quite serious. We are worried," Nice chief executive Julien Fournier said in quotes reported by L'Equipe.
"I am in touch with the family of Kevin and I look forward to the medical verdict. [We] should have information in the next few hours."
Anin joined Nice from Sochaux on a four-year deal in January 2012 but struggled to nail down a first-team place at the Mediterranean club, making just 13 league appearances last season.
The high point of a frustrating campaign on a personal level came in April when Anin scored a last-gasp goal to clinch a 4-3 victory Toulouse. Nice went on to finish the season in fourth place and claim a Europa League berth.
News of Anin's involvement in the accident was met with a number of messages of support on social media.
Nice team-mate Alexy Bosetti tweeted "support my friend Kevin", while Le Mans goalkeeper Jeremie Janot posted "Big thoughts for Kevin Anin! Courage!"
Nice's Ligue 1 rivals Bordeaux also offered supportive words on the club's official Twitter account, saying: "Disturbing news about Kevin Anin. Our support to the player, his family, relatives and @ogcnice #Courage."