Two suspects in the attack in the Dallas suburb of Garland were shot dead after opening fire at a security guard outside the center.
It was unclear whether the group, which has captured large swaths of territory in Syria and Iraq, had an actual hand in the operation, or whether the two suspects had pledged allegiance to the group and then carried out the attack on their own.
U.S. government sources close to the case said on Monday that investigators were scouring electronic communications sent and received by the dead suspects for evidence of contacts between them and armed groups overseas, most notably ISIL.
Officials have identified the suspects as roommates Elton Simpson and Nadir Soofi of Phoenix. Court documents show Simpson had been under surveillance since 2006 and was convicted in 2011 of lying to FBI agents over his desire to join armed groups in Somalia.
In a statement released late Monday by Phoenix law firm Osborn Maledon, Simpson's family said it is "struggling to understand" what happened.
"We are sure many people in this country are curious to know if we had any idea of Elton's plans," the statement said. "To that we say, without question, we did not."
The shooting in Garland was an echo of attacks or threats in other Western countries against images depicting the Prophet Muhammad. In Islam, any physical depiction of the prophet — even a respectful one — is considered blasphemous.
In January, gunmen killed 12 people in the Paris offices of French satirical magazine Charlie Hebdo in what was said to be revenge for its offensive cartoons.
The Texas incident unfolded on Sunday when a car drove up behind the center in Garland, where 200 people were attending an event featuring caricatures of the prophet.
Two men jumped from the car and fired at a police car that was blocking an arena parking lot entrance. A Garland police officer and an unarmed security guard were in the squad car and began to exit as the vehicle approached. The gunmen wounded the security guard, and the police officer returned fire, killing both assailants.
Police and federal agents had planned security for months ahead of the event, organized by American Freedom Defense Initiative (AFDI), an organization described by the Southern Poverty Law Center as a hate group.
The event, “Muhammad Art Exhibit and Contest,” offered a $10,000 prize for the best artwork or cartoon depicting the prophet. Artist Bosch Fawstin won for a depiction of a sword-wielding prophet in a turban shouting, “You can't draw me.”
The AFDI has, among other activities, sponsored anti-Islam advertising campaigns in transit systems across the country.
Al Jazeera and wire services