JetBlue is disputing family members' characterization of an episode that prompted their removal from a June 21 flight and said it is considering barring the family from flying with the airline.

Through a press release from their attorney, Tamir Raanan and Mandy Ifrah said they were returning home to New York from Fort Lauderdale, Florida, with their three children when their 1-year-old, Eden, began kicking a seat in front of them as the plane was about to depart.

The statement said Ifrah and the female passenger in the seat "exchanged a few words" before the woman moved to an empty seat next to her. According to the family, Ifrah apologized to the passenger, but a flight attendant nevertheless asked the pilot to return to the gate, and the family was removed from the flight.

But in a statement to ABC News, JetBlue said the encounter was more serious and "included physical threats and profanities against a nearby customer."

JetBlue said a crew member asked the family to step off the plane to discuss what happened. After they refused "repeated requests," the crew deplaned all passengers, and law enforcement officers escorted the family from the gate area, according to the airline.

A video posted to YouTube shows a flight attendant asking the parents several times to step off the plane to discuss the matter. Both parents refuse. "I'm not getting off this plane," Ifrah is heard saying in the video.

Another video posted to YouTube shows the two arguing with JetBlue staffers and law enforcement in the gate area about their removal. In the video, they demand to know if a woman on the plane is also being removed. It is unclear to whom they are referring.

Raanan and Ifrah said the flight later departed for New York with their luggage, including their clothes and baby supplies, leaving them stranded without many of their belongings overnight. When they returned to the airport the next day, a JetBlue representative said the family was "banned from all future flights and could not provide an explanation," according to the family's press release. It took Raanan and Ifrah a week to get all their luggage, the press release said.

The two claim that JetBlue never informed them why they were kicked off the flight or banned from future flights with the airline.

The lawyer for the family disputed JetBlue's accusation that a member of the family "used profanity and threats of violence."

JetBlue said the passengers were removed because of the parents' actions and it is investigating whether the behavior warrants banning them from future flights.

The family's attorney called JetBlue's actions "unprofessional and reckless, if not malicious." The family has not filed a lawsuit against the airline, as of Wednesday morning.