Images from surveillance cameras in the tunnel at the Parc des Princes, obtained by several local media outlets, appeared to show Bastia's Brazilian forward Brandao waiting for Motta to approach him before headbutting the Brazil-born Italy international.
A mass altercation then ensued, and PSG president Nasser Al-Khelaifi told beIN Sports that Motta's nose had been fractured in the incident.
Al-Khelaifi also called for serious action to be taken against Brandao, the 34-year-old former Marseille player.
"I hope that the Bastia player who did that is very severely punished. A heavy punishment is required," he said.
Meanwhile, PSG's Brazilian captain Thiago Silva, who is currently sidelined with a hamstring injury, used his Instagram account to attack Brandao.
"I want to know how many matches this guy will be suspended for...a guy who is always disloyal!! Who always wants to talk instead of play," wrote Silva, who in May last year was suspended for shoving a referee during a Ligue 1 match against Valenciennes.
"If I got three matches two years ago for touching the referee...Imagine an assault like that?? The fourth official saw everything, I was by his side when this guy attacked our player."
PSG coach Laurent Blanc also accused Brandao of being responsible for the injury that forced full-back Gregory Van der Wiel to come off at half-time of the game, which the reigning champions won 2-0.
Zlatan Ibrahimovic came off injured, after apparently straining a muscle in his side, after just 14 minutes of the match against the Corsicans, before goals from Lucas and Edinson Cavani gave PSG their first Ligue 1 win of the campaign.