The northern side pulled off a remarkable championship success by pipping big-spending Paris Saint-Germain, with Galtier a man in demand following that achievement.
Lyon, Nice and Napoli are said to have made contact with the 54-year-old former Saint-Etienne boss, who could have his choice of jobs.
Olivier Letang, the Lille president, said he gave his blessing for Galtier to leave after being unable to talk him around.
But given that Galtier had 12 months remaining on his Lille deal, the club have a claim on him.
"There's still a year on Christophe's contract with LOSC and in the first instance we will have to discuss the terms of his departure with the club that he chooses," Letang told Lille's website.
"While waiting to find an agreement with this club, we are going to calmly begin to discuss with several coaches of a high level, always with the best interests of the club, namely sporting ambition and competitiveness. Christophe has said he'll always remain a supporter of LOSC; in the same way and in all friendship, I wish him a lot of happiness for the rest of his brilliant coaching career."
Lille clinched the French league title on Sunday and it was Tuesday when Galtier made his desire to leave formally known.
"I obviously showed him my willingness to continue with him, because he is someone for whom I have a great deal of friendship and esteem, as a coach but also as a man," Letang said.
"We spent five months together. They were intense and formidable, on a human and professional level. But Christophe made the decision to leave LOSC. I respect him and have to accept it.
"I naturally tried to convince him [to stay], but I think his decision had already been taken. Everyone at Lille will keep great memories of his time at LOSC, with the peak of this title of champions of France. We can only thank him for what he brought here."