Rather conveniently, producer Stuart Cohen has a blog devoted to his experience while working on The Thing, and it includes an entry about the score.
In a perfect world, given unlimited time and resources, I think John would have preferred to compose the music for THE THING himself. The realities of the work yet to be done, however, combined with the need for a more expansive and layered approach to the score led us to consider other options. We initially offered the film to Jerry Goldsmith who was unavailable, doing both POLTERGEIST and TWILIGHT ZONE for Spielberg. Availability on musician John Corigliano (ALTERED STATES) was checked. The legendary Alex North read the script, had ideas, and wanted to meet but at that point I felt the only composer John would possibly entrust his film to other than himself was Ennio Morricone.
The film was far from complete or coherent - John was still filming in Stewart, so the film lacked most of the exterior scenes as well as amost all of the special effects, save the kennel . Morricone complained about the lack of continuity ( normally we wouldn't have run a film for any composer in this shape, and with the director not present ,but we did not have the luxury of time - we needed to secure his commitment, and were trying to wedge ourselves in to his schedule ) but agreed that if we were to come to him in Rome he would "see what he could do".
Doubts were definitively dispelled two months later when Morricone opened up his tattered valise and removed a reel of two inch tape containing the now-emblematic "heartbeat " theme. As we heard this for the first time in the recording booth at Universal I looked over at John, whose expression was initially one of relief, followed by something close to wonder... it seemed that Morricone had understood John perfectly. At the orchestral recording session the next day, I remember John coming in late and shyly taking a seat in the back, an observer for the first time as Morricone recorded the rest of the music for his movie. Having been recorded in large brushstrokes of sound, there was still the need for more specific transition and suspense cues which John, along with his partner, Alan Howarth, then supplied.
So basically, Carpenter already had too much on his plate, and was forced to turn over responsibility for the score to an outside composer. The only person Carpenter trusted with his baby was Ennio Morricone. In the end, everyone walked away happy.