Marine tracking shows the tanker moving east into the Mediterranean and lists Kalamata in Greece as the destination.
Gibraltar earlier rejected a request by the US to again seize the ship, which has changed its name from Grace 1 to Adrian Darya-1.
The US made the last-minute request on Friday, a day after Gibraltar lifted its detention order.
Gibraltar said it could not comply with Washington's request to issue a new detention order because US sanctions against Iran did not apply in the EU.
Tehran said it was ready to dispatch a naval escort to the Adrian Darya-1.
A British-flagged tanker seized in July remains in Iranian hands. There had been speculation of a swap if Grace 1 was freed, despite official denials.
A spokesman for the Iranian foreign ministry said on Monday 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.
What's the background?
The ship with its crew of 29 - from India, Russia, Latvia and the Philippines - was seized with the help of British marines on 4 July, after the government of Gibraltar - a British territory - suggested it was heading for Syria in breach of EU sanctions.
The move sparked a diplomatic crisis between the UK and Iran, which has escalated over recent weeks and saw Iran seize a British-flagged and Swedish-owned oil tanker, Stena Impero, in the Gulf.
The Gibraltar authorities freed the vessel on Thursday after receiving assurances from Iran that it would not discharge its cargo in Syria.
Why was the US request denied?
Gibraltar, in a statement on Sunday, said it could not comply with the request because Iran's Islamic Revolutionary Guard Corps is not viewed as a terrorist organisation by the EU, which the British territory is currently part of.
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".