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 other countries affected are Iran, Iraq, Libya, Somalia, Sudan and Yemen.
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.
Mark Doss, a lawyer for the two Iraqis, spoke to the BBC from Terminal 4 at JFK, where he has been since Friday evening trying, and failing, to see his clients:
“It’s just absolutely disgraceful that we would be turning back and detaining the most vulnerable individuals in the world – individuals who have served our country, are fleeing from persecution and have valid status, only to be detained without an attorney at their port of entry [to the US].
“And so we believe this order is unconstitutional and we will be fighting it.”
Several other rights groups, including the American Civil Liberties Union (ACLU), are involved in the lawsuit, filed 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. (BBC)