Naval tensions have increased dramatically since a series of incidents in 2019.
Iran has denied allegations that the seizure of the tanker and its 20-member crew amounted to hostage taking, and said it was Seoul that was holding Iran's funds "hostage."
US officials said over the weekend they have a “high degree of certainty” that Maduro has been paying Iran in gold for the fuel.
A spokesman for the ship's owner, Stena Bulk of Sweden, said on Friday that the vessel was preparing to leave.
The blacklisted tanker, which is loaded with Iranian crude oil, sent the last signal giving its position between Cyprus and Syria sailing north at 15:53 GMT on Monday, Refinitiv data showed.
“The vessel is now headed to Dubai, where it will refuel before beginning a months-long journey around the horn of Africa, through the Mediterranean and to the shores of Syria.”
The Grace 1 oil tanker was seized by Royal Marine commandos in a landing in darkness off the coast of the British overseas territory on July 4.
Britain rejected the idea that it could release a Iranian tanker in exchange for a British-flagged vessel seized by the Islamic Republic in the Gulf.
Iran has already seized a UK tanker in response and London has said it will attempt to protect its ships from more Iranian harassment.
Also on Sunday, Iraqi Prime Minister Adel Abdul Mahdi has ordered an investigation into a strike against a Popular Mobilization Force’s (PMF) base this past weekend.