It depends what you mean by "city-state". If you are referring to Emirates (Arab Tribal Principalities), then the largest in area is Kuwait, which sits on the Persian Gulf.
If you are talking about historic city-states, there are several historic city-states in the Mesopotamian valley, like Akkad or Sumer. Additionally, as the Persian Gulf lies between modern day Saudi Arabia and Iran, the closest Ancient Greek city-state would have been Knossos, where the Mycenaean Greeks lived.
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There are actually several small countries (which are small enough to qualify as city-states) located directly IN the Persian Gulf:
Qatar
The first is probably the best fit for a definition of both "city-state", and "closest to", being that it is barely 750 sq km in size, and effectively is an island archipelago off the shore of Saudi Arabia.