In Cairo, Egypt, two sources close to airport security told NBC News that an Iraqi family of five from the city of Erbil had been prevented from boarding a flight to New York.
NBC News was unable to immediately confirm whether this was related to the travel restrictions. A United Nations refugee agency spokesperson referred questions on the Iraqi family to U.S. authorities.
Trump's executive order suspended admission of all refugees to the U.S. for 120 days. It also indefinitely bans refugees from Syria and restricts entry to the country for people from six other predominantly-Muslim nations for 90 days.
In signing the order, Trump pledged to "keep radical Islamic terrorists out of the United States of America."
The State Department and Customs and Border Patrol on Saturday were still devising the formal rules for who is eligible for waivers — such rules were not written up before the executive order was implemented.
The Trump administration also has yet to issue guidance to airports and airlines on how to implement the executive order. "Nobody has any idea what is going on," a senior Homeland Security official told NBC News.
In addition, the order applies to both refugees who are still "in transit" and U.S. green card holders from the affected Muslim-majority countries currently overseas, a senior administration official said. Those people would also need to apply for a waiver if they want to return to America, the official added.
The official said it would have been "reckless and irresponsible" to issue guidelines before the executive order was signed because of security reasons.
The International Rescue Committee called the decision to halt the U.S. refugee resettlement program a "harmful and hasty" decision.
Meanwhile, the United Nations refugee agency and International Organization for Migration called on the Trump administration to continue offering asylum to people fleeing war and persecution, saying its resettlement program was vital.
In Paris on Saturday, French Minister of Foreign Affairs Jean-Marc Ayrault said Trump's orders "can only worry us."
"Welcoming refugees who flee war and oppression is part of our duty," Ayrault said in comments carried by Reuters.
Qatar Airways issued a statement on its website that said nationals from Syria, Iraq, Iran, Sudan, Somalia, Yemen and Libya may only travel to the U.S. if they are in possession of a resident green card or specific visas usually granted to government employees, individuals traveling to the United Nations or employees of international organizations.