There are several synonyms for the phrase "come close." One such synonym is "approach," which refers to moving nearer to something or someone. Another synonym is "nearly," which means almost or very close to achieving something. "Near" is also a synonym for "come close" and denotes proximity. The phrase "draw near" can also be used to describe coming close to someone or something. Lastly, "mimic" or "emulate" can be used to describe a situation where one tries to imitate something very closely, as if coming close to replicating it. With these synonyms, one can communicate the idea of coming close in various contexts.