Santos, 24, suffered a ruptured anterior cruciate ligament and meniscal tear in his right knee and was taken off the field on a stretcher after being tackled by Le Havre midfielder Amir Richardson during the first half of Lille's 1-0 victory.
The referee gave a foul for the tackle but did not issue Richardson with a caution.
Lille said the tackle merited a red card and that they had asked the league's governing body (LFP) to refer the matter to its disciplinary committee. The club are also seeking an explanation of the video review process undertaken by match officials.
"LOSC is exploring all possible legal avenues and reserves the right to take any legal action necessary to obtain compensation for its losses," the club said. "It will support Tiago in any steps he decides to take to defend his rights."
In response, Le Havre wished Santos a quick recovery, but highlighted that neither the match referee nor the Video Assistant Referee deemed the tackle worthy of punishment.
"Given this, HAC questions LOSC's move to have the FFF (French federation) and the LFP punish its player after the fact, since such a decision would fall outside the current regulations," the club added.
"HAC respects all fellow Ligue 1 clubs and expects the same respect in return."
The FFF and LFP did not immediately respond to a request for comment outside regular business hours in France.
Portuguese defender Santos previously suffered a ruptured ACL in his left knee in 2024.
Lille, second in Ligue 1, face Arsenal in London in the Champions League later on Tuesday.