The France Under-20 midfielder has missed training in an attempt to force a move away from the Stade Pierre-Mauroy, with Marseille and Newcastle keen on his services
The talented 20-year-old was reported to have been absent from training on Tuesday as he attempts to force a move away from les Dogues.
Marseille have already had two separate bids for the former Bastia player rejected, with Lille president Michel Seydoux adamant the player will remain at the club.
"If we give in to whatever kind of pressure, the power wouldn't be in the right place," Seydoux told RMC.
"There’s no reason for him to leave Lille. I don’t have a crystal ball, but I am 99.9% sure he will stay."
Thauvin signed for the three-time Ligue 1 champions from Bastia in January, but was loaned back to the Corsican outfit for the remainder of the 2012-13 season.
The former Grenoble youngster was an important player in France’s Under-20 World Cup win during the summer, and his displays in Turkey alerted a host of top European clubs.
Marseille had an initial offer of €5.5m [£4.7m] flatly rejected by Lille in June, with the Stade Velodrome club having another bid worth €10m [£8.6m] rejected last week.
Lille’s refusal to countenance Thauvin’s departure has only strengthened his intention to leave, with his refusal to train on Tuesday being seen as a signal of intent to force through a move.
Thauvin’s actions have been criticised by Seydoux, who believes the youngster is only "hurting himself".
"The real victim is Florian Thauvin, who's not playing football," he added.
“He's making his own problems. I think his advisors are not the best in the world. It should be very simple, and it's a bit too complicated for me.
"Lille lived before Florian Thauvin and will live after him. The question that needs asking is why the advice he's getting is so bad.
"We're in a normal world, he's not being exploited, he's in a rather good club with good ambitions. I think Florian has everything to gain in closing this chapter and putting on his boots."
L’Equipe have reported Lille have asked a Belgian agent to try to reconcile with Thauvin or find the player a new club.
The sports daily also suggested the 2010-11 double winners were softening their stance on the player previously described as "untransferable" and were privately prepared to accept a bid of €15m to wash their hands of Thauvin.
Marseille remain in pole position to win the race for his signature, but Newcastle may also renew their interest after missing out in January.