OM have been forced to fend off interest in the 30-year-old playmaker with Arsenal reportedly keen on agreeing a season-long loan deal.
The Gunners are searching for midfield replacements following Cesc Fabregas' departure for Barcelona, while Samir Nasri is expected to move to Manchester City.
Any interest in Gonzalez would apparently seem likely to fail, however, after Anigo told L'Equipe: "It's not even a debate. We do not want him (Gonzalez) to leave.
"The transfer window closes in less than 15 days and I don't see how that would be enough time to find a suitable replacement."
Anigo is, however, open to beginning negotiations with Arsenal over a possible deal for France international Mathieu Valbuena.
OM are yet to receive any offers for Valbuena, who has eight France caps, but Anigo said the club were willing to let him go rather than fellow midfielder Andre Ayew.
"Mathieu has been with this club for six years and we're willing to open talks if Arsenal want to buy him," he told RMC.
"We are very pleased with his performances, but we're ready to listen to offers out of respect for the player.
"However, we have not had contact with Arsenal. Not about Mathieu and not about Andre [Ayew] either."
Anigo ruled out any sale of Ghana international Ayew, who was born in the northern France town of Seclin, adding: "Andre Ayew is not for sale.
"It is correct that he has a clause in his contract, but that clause was valid until a certain date and that day has already passed.
"If clubs want to buy him now, they'd have to pay another fee than that clause. Andre still has plenty to achieve at Marseille."