My guess is spirits of the Cetra or some form of the black cloaked Reunion people. The Reunion people were injected with Jenova cells and are trying to reconverge after the Nibelheim event which happened in a flashback. The wiki says that Sephiroth initiated the Reunion 5 years after the Nibelheim event when Cloud and Tifa met up again (the beginning of the game more or less), which pre-dates him meeting Aeris and would explain those being the Reunion people.
From the wiki:
"The survivors of Nibelheim, including Cloud Strife and Zack Fair (and excluding Tifa Lockhart, who had been rescued before Shinra arrived), were rounded up and injected with Sephiroth cells, Sephiroth having merged with Jenova cells while still in the womb. The subjects were exposed to Mako in a procedure similar to the augmentation done to SOLDIER members. As the Nibelheim survivors did not have the mental or physical strength to keep their sense of self, they became mindless puppets devoted to Jenova and Sephiroth called Sephiroth Clones."
If they have the instinct that Aeris is a threat to Jenova's life now that she has met up with someone who is strong(ish) enough to resist the draw of the Reunion and is also fighting against that and Jenova's goal of converging, it would make sense they would be going after her/them. They showed the ghastly forms that some people ended up in after being exposed to Mako during the Nibelheim scene. I don't think it's too far-fetched to say that some people could have turned into those things.