The Adrian Darya 1’s plan to unload its oil cargo and collect fresh supplies in Greece is hitting a snag, people familiar with the tanker’s operations said, as the U.S. pressures Athens to avoid any dealings with the Iranian vessel.

The ship, previously called the Grace 1, left Gibraltar on Sunday after the British overseas territory rejected a U.S. request to seize the ship. The Greek government is reluctant to host the ship in its waters, a senior European diplomat said.

The...