I don't know for sure, but the three theories are: Zack, Jenova, and Cloud's inner self.
I've always assumed it was Cloud's inner self, as "Back then, I only got scraped knees" would make most sense in the context of Cloud and Tifa falling off of the bridge at Mt. Nibel, with Tifa being critically injured and Cloud only scraping his knees.
Of course, with the introduction of all this Crisis Core bullshit, people are starting to draw the connection to how Zack fell and landed in the exact same place (from the opposite side of Midgar
) and met Aerith in the past, and use it as evidence that it's Zack's spirit talking to Cloud.
To me, these are both perfectly valid explanations, and hell, it could even be that Zack and Cloud's consciousnesses merged somehow after Hojo's experimentation or something, in which case it'd be both, and it'd make sense how Cloud stopped hearing the voices after he got a grasp of who he was and separated his consciousness from Zack's.
Claiming it was Jenova makes no sense to me, unless she's just trying to confuse him...but even that wouldn't really make sense.