Which of the two Talmuds would the 7th century Jews have been made perverse to?

The Balvi is the common text used now from the Babylonian Jewish Rabbis. But it seems the Yerushalmi may have been the original, despite being uncompleted.

My question being, we know that the Jews were scattered during the Roman Empire and suppressed Jewish school of thought. So quite a few fled east and towards safer lands. But which book did they have with them at the time of Prophet Muhammad?