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.”
The Hollywood Reporter said 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. I would like some privacy and time to heal. 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. Love Sylvia.”
A Ventura County fire spokesman confirmed that a plane registered to Horner had crashed, that the pilot had died and that no one else was on board.
A lawyer for Horner, Jay Cooper, said that the aircraft was one of many owned by the composer.
Stars across the globe have been remembering Horner, who had 10 Academy Award nominations, including two wins for Titanic.
Horner also famously worked on Braveheart, and actor Russell Crowe paid tribute to him, writing: “My sincere condolences to the family, loved ones and friends of James Horner. #abeautifulmind”.
Horner, 61, won two Academy Awards for his work on Titanic, one for the score and one shared with lyricist Will Jennings for best original song ‘My Heart Will Go On’, performed by Celine Dion.
Horner also composed the music for Aliens, The Karate Kid and a string of other major films. His scores for Avatar, A Beautiful Mind and House of Sand and Fog earned Oscar nominations.
Horner was also behind music for Apollo 13, and Field of Dreams. His first nomination came in 1986 for the sci-fi horror film Aliens starring Sigourney Weaver.
American film touchstones of the 1990s such as Patriot Games, Searching for Bobby Fischer and Jumanji were also composed by Horner.
Top actors and directors in the film and television world paid tribute to Horner online.
“Incredibly saddened to hear about the loss of James Horner. I grew up loving his work. He leaves behind a spectacular musical legacy,” television series creator Seth MacFarlane tweeted.
“There is nothing that shaped my movie-going experience more than the musical genius of James Horner. He will live on through the ages,” tweeted actor Rob Lowe.
Actor Ron Perlman, who starred in Enemy at the Gates, said he was “lucky to be able to say I was in a movie that was scored by James Horner.”
“Flights of angels dear sweet beautiful mind...,” he wrote on Twitter.
The plane was an S-312 Tucano MK1 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, 100 miles northwest of Los Angeles.