Adrian Darya 1, formerly known as the Grace 1, slowed to a near-stop on Sunday some 50 nautical miles (92 kilometers) off Syria, the ship-tracking website MarineTraffic.com showed on Sunday.
The ship still does not list a destination for its 2.1 million barrels of oil, worth some $130 million.
The Adrian Darya was held for weeks off Gibraltar after being seized by authorities there on suspicion of violating EU sanctions on oil sales to Syria.
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The U.S. has a warrant in federal court to seize the ship and has been warning nations not to accept it.
Meanwhile, Iran's deputy foreign minister and economists are to travel to Paris on Monday to speak with French officials.
On Friday, the Wall Street Journal reported that the tanker could still be looking to bring its cargo to Syria, in violation of US sanctions and an Iranian promise made to British authorities.
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Quoting sources familiar with the ship’s plans, the report said it was looking to split its load of crude oil onto smaller ships, which would then be able to complete the cargo’s suspected journey.
The U.S. Treasury Department announced it had blacklisted the tanker and sanctioned its captain, while U.S. Secretary of State Mike Pompeo said Washington believed the tanker was headed to Syria.
"Anyone providing support to the Adrian Darya 1 risks being sanctioned," Treasury Under Secretary Sigal Mandelker said in a statement.
"We have reliable information that the tanker is underway and headed to Tartus, Syria. I hope it changes course," U.S. Secretary of State Mike Pompeo said on Twitter.