"These attacks are serious and go beyond a simple verbal clash," the Ligue 1 leaders said in a statement.
"Thiago Silva and PSG reserve their right to take any necessary action."
Barton, who has more than two million followers on Twitter, used the social network to insult the Brazilian with messages that also drew condemnation from gay activists.
London-based gay football team Stonewall FC tweeted in response: "Transphobia in sport is just as bad as homophobia in sport - @joeybarton's comments this morning need looking at."
Barton has been involved in an ongoing spat with Thiago Silva this week following disparaging comments the Briton had made about some of the PSG player's international team-mates.
Marseille coach Elie Baup told a news conference earlier on Wednesday that his responsibility did not extend further than the pitch, adding that Barton was responsible for his own comments on social media.
The French federation said that its Ethics Committee would not take action because the incident did not happen on a football pitch.
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