According to reports, Mr Horner was killed when his two-seater single-engine plane crashed in southern California.
Howard wrote on Twitter: “Brilliant composer James Horner, friend & collaborator on 7 movies, has tragically died in a plane crash. My heart aches for his loved ones.”
Mr Horner’s personal assistant, Sylvia Patrycja, confirmed the news on her Facebook page.
She wrote: “A great tragedy has struck my family today, and I will not be around for a while.
“We have lost an amazing person with a huge heart and unbelievable talent. He died doing what he loved. Thank you for all your support and love and see you down the road.”
A Ventura County fire spokesman confirmed that a plane registered to Mr Horner had crashed, that the pilot had died and no-one else was on board.
A lawyer for Mr Horner, Jay Cooper, said the aircraft was one of many owned by the composer. He added that no-one had heard from Mr Horner since the crash. Mr Cooper said: “It was his plane and, if he wasn’t in it, he would’ve called.”
Stars across the globe have been remembering Mr Horner, who had ten Academy Award nominations, and won two Oscars for Titanic.
He also worked on Braveheart, while actor Russell Crowe paid tribute, writing: “My sincere condolences to the family, loved ones and friends of James Horner. #abeautifulmind”.
American composer Steve Jablonsky, best known for his music in the Transformers movie series, tweeted: “James Horner, thank you for inspiring a young me to pursue a career in film music. Thank you for your music. Rest in peace.”
The Amazing Spider-Man director Marc Webb also paid tribute, writing on Twitter: “Last time I spoke to JH he was scoring for a kid. Two Oscars and he agreed to score a student film! What generosity.”
The plane was a Tucano turbo-prop with two seats, said Ian Gregor, spokesman for the Federal Aviation Administration.
Crews extinguished a fire that erupted in vegetation surrounding the remote crash site, about 100 miles from Los Angeles.
Mr Horner’s Academy Award wins for Titanic were for the film’s score and its enduring theme song, My Heart Will Go On, sung by Celine Dion.
His scores for Alien, Apollo 13, Field of Dreams, Braveheart, A Beautiful Mind and Avatar also earned Oscar nods.