The car, a 1995 Camry which belongs to a member of the Islamic Center of Lafayette, was set on fire near the trunk but didn't suffer much damage, fire officials told The Advocate.
The fire department, police department, Lafayette Parish Sheriff's Office, and the local FBI are investigating the crime.
"We're looking into a possible civil rights violation involving a possible hate crime," Special Agent Sheila Thorne told the Advocate.
The president of the Islamic center, Hesham Tellawi, told the Advertiser it was too soon to call it a hate crime.
"It could not be a hate crime and be just a normal crime," Tellawi said. "We don't want to say it is when it could very well not be. But if it was, we would look at it differently."
Late update: A spokesman for the Lafayette Police Department, which initially responded to the fire and is investigating, said it's the first such incident at the Islamic center.
"We haven't had any issues, such as protests, ever," said Cpl. Paul Mouton. He said they've had no reports of any vandalism or defacing of property, either.