"We have lost an amazing person with a huge heart and unbelievable talent," wrote Horner's personal assistant Sylvia Patrycja on Facebook.
"He died doing what he loved. Thank you for all your support and love and see you down the road."
The Federal Aviation Administration says the plane was an S-312 Tucano MK1 turbo-prop with two seats.
Horner, 61, won two Oscars for his work on the 1997 film Titanic – one for the score and another for its theme song.
He was nominated for a further eight Oscars for films Avatar, House of Sand and Fog, A Beautiful Mind, Apollo 13, Braveheart, Field of Dreams, An American Tail and Aliens.
On hearing about Horner's death, Russell Crowe added: "My sincere condolences to the family, loved ones and friends of James Horner."
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