Barton suggested that Silva was a transsexual on Wednesday after the defender had taken exception to criticism of the Brazilian national side.
"These last hours on social networks, unacceptable statements were made by Joey Barton, player of Olympique Marseille against Paris Saint-Germain defender Thiago Silva," read a PSG statement.
"These attacks are serious and go beyond mere verbal joust. The officers of the club, the players and the technical staff, supporting Thiago Silva cannot accept such a statement.
"They reiterate their firm commitment to the values of respect that should be the rule of our sport. Thiago Silva and Paris Saint-Germain reserve the right to take any action they deem necessary."
Barton is currently on loan at Marseille from Queens Park Rangers until the end of the season.