The US has accused the Marinera, formerly known as the Bella 1, of breaching sanctions by carrying oil for Venezuela, Russia and Iran.
The UK government backed the operation to seize the tanker, saying it was a lawful action against a vessel involved in breaking sanctions.
The Ministry of Defence (MoD) provided operational support and US planes used UK airfields, including some in Scotland.
But the Russian government denounced the seizure, demanding that the US treat Russian nationals aboard properly and allow them to return to Russia quickly.
In Wednesday’s statement, the US Embassy said the “United States and the United Kingdom are coordinating regarding the status, processing, and prospective repatriation of crew members from the Bella 1.”
It said the vessel had been “interdicted” in international waters and moved into UK territorial waters “for safety reasons.”
The statement added that “US and UK officials across several agencies are working together closely to determine a way forward for final disposition of the ship.”
At Holyrood, Swinney also called for “proper lines of communication” with the UK government, after saying ministers in Edinburgh had been left in the dark about the US-led operation.
He said his government was not told in advance about the captain and first officer being taken into US custody.
He insisted that Scottish ministers and officials had followed proper legal processes.
Asked by Labour MSP Neil Bibby if he had received security briefings from the UK government, the first minister confirmed they had been provided.
The UK government has previously defended its dealings with Swinney’s government, with Scottish Secretary Douglas Alexander saying it was “very clear” that defence and foreign affairs were under the control of Westminster not Holyrood.