Rights groups have filed a lawsuit in a New York court to demand the release of refugees in transit who have been detained at John F Kennedy airport.
Entry to the US for nationals of seven Muslim-majority countries has been stopped for 90 days by Donald Trump.
The exact implications of his order remain unclear. The US State Department has told the BBC it is working on the immediate implementation of the ban.
People fleeing Syria are banned until further notice.
The two Iraqi refugees detained in New York, one of whom had worked as a US Army interpreter, were in transit when the executive order was signed on Friday.
The National Immigration Law Centre (NILC) told the BBC that it was suing President Trump and the secretary of the Department of Homeland Security.
It described the two Iraqis as “courageous Haneed Khalid Darweesh, who interpreted for US army & Haider Sameer Alshawi also targeted for aiding US military”.
The organisation had been unable to speak to the two men, it said.
The American Civil Liberties Union (ACLU) is one of several other rights groups also involved in the lawsuit, filed in the Eastern District of New York on Saturday morning.
Lawyers are “keeping tabs on several flights” the NILC told the BBC, but did not have a full number of people who had been detained at US airports.
On Saturday several Iraqi passengers and a Yemeni national were prevented from boarding a flight at Cairo airport bound for New York, despite holding valid visas for the US.
Google has urged travelling staff members who are nationals of the seven countries affect to return to the US as quickly as possible.
Friday’s wide-ranging order includes the following measures:
- The suspension of the entire US refugee admissions programme for 120 days
- A ban on all refugees from Syria until “significant changes” are made
- A 90-day suspension on anyone arriving from Iraq, Syria, Iran, Libya, Somalia, Sudan, and Yemen, except certain visa categories such as diplomats
- Priority for future refugee applications from those persecuted for their religion – but only if the person is part of a minority religion in their home country
- A cap of 50,000 refugees in 2017 – less than half of the upper limit under Mr Trump’s predecessor, Barack Obama
Mr Trump signed the order on Friday, which was International Holocaust Remembrance Day.
The president’s statement to mark that occasion, on the 72nd anniversary of the liberation of Auschwitz, made no mention of Jews or anti-Semitism.
In response to Mr Trump’s order, the United Nations refugee agency said the needs of those fleeing conflict had never been greater.
The Council on American Islamic Relations (CAIR) also says it will file a lawsuit.
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