Could slaves become citizens in Athens and Rome?

Yes they could have became citizens of Rome and Athens because the law changed sooner than later.

Explanation:

That law was relaxed as well as time went on; for example, children of freed slaves could apply to become citizens. Even if both parents were Roman citizens, children had no rights. Boys of Roman citizens went though a ceremony when they were 16 or 17, depending upon how close their birthday was to March 17th, and at that time became citizens of Rome with full benefits.