Marine tracking has showed the tanker, which was released on Sunday, moving east into the Mediterranean and lists Kalamata in Greece as the destination, the BBC said in its report.
The ship with its 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 British territory, suggested it was heading for Syria in breach of European Union (EU) sanctions.
The 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.
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.
It also said that US sanctions preventing oil exports from Iran could not be enforced by the EU, reflecting what it said were "the very different positions and legal regimes in the US and the EU".
Iran's Ambassador to Britain, Hamid Baeidinejad, said in a tweet earlier on Sunday: "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 Sunday'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.
( With inputs from IANS )