The attacker has been in excellent form since the start of this season.
But he is out of contract at the end of the campaign, and could sign with other clubs for free.
He can talk to teams in England starting in January, and Leicester are interested, per Leicestershire Live.
Marseille sporting director Pablo Longoria admitted no agreement has been found.
"Thauvin is a club heritage, that's why it's hard to lose an important player like him," said Longoria, reported telefootlachaine.fr.
"He will be free to agree terms with another club [in January] and that's why we all have to work together – club, player, entourage – to find the best solution."