The teenager had last been seen entering the bathroom on Thursday and had a history of seizures, according to the statement. An autopsy is planned.
Family attorney Michael Ossi said in a statement that Jett died suddenly on Friday. Publicists Samantha Mast and Paul Block released the statement but could not be reached for additional comment.
A local police source said Jett apparently hit his head on the bathtub.
Jett was the oldest child of Travolta, 54, and his wife, actress Kelly Preston, 46, who also have an 8-year-old daughter, Ella Bleu.
Preston has said that Jett became very sick when he was 2 years old and was diagnosed with Kawasaki disease, an illness that leads to inflammation of the blood vessels in young children. She blamed household cleaners and fertilizers, and said that a detoxification program based on teachings from the Church of Scientology helped improve his health, according to People magazine.
It is unclear whether Jett was taking any medications for his seizures.
The Scientology Celebrity Center in Los Angeles declined to comment. Both Travolta and Preston are practicing Scientologists.
A spokeswoman for Rand Memorial Hospital in Freeport said she could not release any information because of privacy concerns.
The family had arrived in the Bahamas on a private plane Tuesday and was vacationing at their home in the Old Bahama Bay resort community.
"The Travolta family has become like family to us at Old Bahama Bay and we extend our deepest sympathies to them," said Robert Gidel, president of Ginn Resorts, the property's owner.
Family friend and Tampa-based attorney Michael McDermott told the Ocala Star-Banner on Friday that Travolta was "broken up" and noted that "Kelly's very quiet."
"Just say a prayer for them," he added. "That's all you can do at this point."
McDermott said that one of Travolta's close associates was making funeral arrangements in Ocala, Florida, where Travolta has a home.
Travolta, who gained fame as Vinnie Barbarino on the 1970s television show "Welcome Back, Kotter" and the 1977 film "Saturday Night Fever," went on to become one of Hollywood's biggest names. He married Preston in 1991.
A television actress, Preston appeared with Travolta in the 2000 film "Battlefield Earth," based on a novel by Scientology founder L. Ron Hubbard.
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Associated Press Writer Josh Dickey in Los Angeles contributed to this report.
