The MarineTraffic website has showed the tanker, which was released on Sunday, moving east into the Mediterranean and lists Kalamata in Greece as the destination, the BBC reported.
The ship, with its cargo of 2.1 million barrels of oil, and crew of 29 from India, Russia, Latvia and the Philippines, was seized with the help of British marines on July 4, after the government of Gibraltar, a UK territory, suggested it was heading for Syria in breach of European Union (EU) sanctions.
Tehran, however, rejected the claim that the tanker was heading to Syria and slammed the seizure as "piracy".
Gibraltar authorities freed the Adrian Darya-1 on August 15 after receiving assurances from Iran that it would not discharge its cargo in Syria.
The next day, the US Justice Department filed a request to detain the ship on the grounds that it had links to Iran's Islamic Revolutionary Guard Corps (IRGC), which it has designated a terrorist group.
But Gibraltar in a statement on Sunday, said it could not comply with the request because the IRGC was not viewed as a terrorist organisation by the EU, which the British territory is currently part of, the BBC reported.
Iran's Ambassador to London Hamid Baeidinejad also, in a tweet on Sunday confirmed that the supertanker had started its journey in international waters 45 days after being seized in Gibraltar.
"Around the clock efforts have been underway to perform the logistical procedures at the port and to secure a complete crew. Contacts with the coordinating company are happening moment by moment.
"With the arrival of two engineering teams from two separate locations, the tanker is expected to depart Gibraltar tonight."
The US is yet to comment on Monday's development.
The seizure of the Adrian Darya-1 sparked a diplomatic crisis between the UK and Iran, which has escalated over recent weeks and saw Tehran seize a British-flagged and Swedish-owned oil tanker, Stena Impero, in the Gulf.
The Stena Impero, which was seized by the IRGC on July 19, remains in Iranian hands.
A spokesman for the Iranian Foreign Ministry said on Monday that they were waiting for a court order before any possible release of the ship but said there was "no connection whatsoever" between the two ship seizures, the BBC reported.
He also warned the US not to attempt to seize the ship in international waters
Regarding the changing of the tanker's name, Iran said last week that it was due to the rejection of Panama, who was the previous flag state of the ship, to continue to keep it registered after it was detained, Press TV reported.
Ambassador Baeidinejad had said that the name change was in line with international maritime rules, while rejecting claims that the move was aimed at avoiding US sanctions.
( With inputs from IANS )