The phrase "swivel eye" is a slang term used to describe someone who is nosy or always looking around in a suspicious manner. There are many other words that can be used as synonyms for this term, including "prying," "nosy," "intrusive," "curious," "meddlesome," and "inquisitive." Other related words might include "snoopy," "busybody," and "gossipy." While these terms may have slightly different connotations, they all relate to the idea of someone who is overly interested in the affairs of others, and who is always watching, listening, or otherwise meddling in things that don't concern them.