Ship tracking data showed the Russian-flagged tanker Anatoly Kolodkin on Sunday was just off the eastern tip of Cuba.
The Times report, citing a US official briefed on the matter, said it was unclear why the US Coast Guard was allowing the shipment to go through. A move by Washington to block the tanker by force, however, could have raised tensions at sea with Russia.
The Coast Guard referred queries on the matter to the White House, which did not immediately reply to a Reuters request for comment.
US President Donald Trump has effectively blocked all oil shipments to Cuba in an attempt to pressure the government in Havana.
Separately, the US temporarily eased sanctions on Russia to help improve the flow of oil that has been restricted by the US-Israeli war with Iran. In the meantime, the Anatoly Kolodkin has been making its way to Cuba.
The vessel departed from Russia's Primorsk port carrying some 650,000 barrels of crude, LSEG ship-monitoring data also showed.
It could soon discharge at Cuba's Matanzas port if it does not change its current course, according to ship tracking site Marine Traffic.
That much oil would provide significant relief to Cuba, which, according to President Miguel Diaz-Canel, has not received any oil imports for three months, leading to strict rationing of gasoline and exacerbating an energy crisis that has resulted in multiple power outages across the communist-ruled nation.
When the US military captured Venezuelan leader Nicolas Maduro on January 3 and whisked him back to the US to face drug-trafficking charges, it removed from power a crucial Cuban ally who had been providing oil to Havana on favourable terms.
The Trump administration blocked all Venezuelan oil shipments to Cuba and threatened to impose punitive tariffs on any third country that supplied the Caribbean island, leading Mexico to halt its exports to Cuba.
Another vessel of Russian origin, the Hong Kong-flagged Sea Horse, had been bound for Cuba with 200,000 barrels of fuel, but it was rerouted to Venezuela.
