The Grace 1 was seized by British Royal Marine commandos in darkness at the western mouth of the Mediterranean on July 4 on suspicion of violating European Union sanctions by taking oil to Syria, a close ally of Iran.
Gibraltar Chief Minister Fabian Picardo decided to lift the detention order after written assurances from Tehran that the ship would not discharge its oil in Syria.
But the fate of the ship has since been complicated by a last-minute legal bid to hold it.
"She is able to leave as soon as she organizes the logistics necessary in order to sail a ship of that size," Picardo told BBC Radio. "Could be today, could be tomorrow."
Asked about the U.S. request, he said: "Those will be determinations made purely objectively and independently by those authorities and then subject to once again the jurisdiction of the Supreme Court of Gibraltar.
"It could go back to the court absolutely."
In a statement, the U.S. State Department said the United States had determined the ship was helping Iran's Islamic Revolutionary Guard Corps, which Washington has designated as a terrorist organization.
U.S. Secretary of State Mike Pompeo took to Twitter to warn all mariners that if they crewed a ship affiliated to the Islamic Revolutionary Guard Corps, they would jeopardize their ability to enter the United States. "#NotWorthIt," he said.
Two weeks after the Grace 1 was held, Iran seized a British- flagged tanker in the Strait of Hormuz leading into the Gulf. That fed into wider hostilities since the United States last year pulled out of an international agreement curbing Iran's nuclear program and reimposed economic sanctions.
Iran said the tanker would sail shortly.
"Based on the owner's request, the oil tanker Grace 1 will depart for the Mediterranean after being reflagged under the Iranian flag and renamed Adrian Darya after preparing for the journey," Iran's state television quoted Jalil Eslami, deputy head of Iran's Ports and Maritime Organisation, as saying.
"The 25-member crew will start their journey after preparations, including refueling," Eslami said.The television report did not identify the tanker's owner.