Here's a quotation from Ifukube Akira 伊福部昭, found in Kobayashi Atsushi's enjoyable if non-technical Gojira no ongaku ゴジラの音楽 ("The music of Godzilla"). It's sourced to a 1995 "statement at Ifukube's residence," so I suppose it was an interview conducted by Kobayashi himself.

At first, the sound crew tried to create it by processing the voices of real animals. They apparently went to a zoo and recorded the voices of lions, tigers, elephants, condors, night herons, and so on, but they couldn't come up with a sound that was it. Nothing was the right voice for Godzilla. It makes sense: mammals have mammalian voices, no matter what; birds will always have avian voices. That was when I suggested using the longitudinal vibration on a contrabass string. Stringed instruments normally use latitudinal vibration and maintain fixed pitches, but we tried rubbed this string lengthwise with a leather glove covered with pine sap instead of a bow. Or to put it another way, we gripped it and pulled. When we did that, the tension changed while the string was sounding, and we got a complex noise like we'd never heard before. We got that down on tape and then Minawa [Ichirô]-san in Sound Effects processed it. And that was Godzilla's voice.

A bit lower down, assistant director Tokoro Kenji 所健二 clarifies: they did add the animal cries on top of the contrabass string. (Also, they had to record them at night, because it was too noisy in Ueno Zoo during the day, but that meant getting the zoo attendants to do things like poke sleeping lions with sticks.)