"We didn't overrule the Department of Homeland Security, as far as green card holders moving forward, it doesn't affect them", Priebus first said.
He did not elaborate on the change.
A Department of Homeland Security official said no green-card holders from the seven countries named in the order had been prevented from entering the US.
PRIEBUS: First of all, it's not affecting green card holders.
The order barred USA border agents from removing anyone who arrived in the US with a valid visa from Iraq, Syria, Iran, Sudan, Libya, Somalia and Yemen. You keep saying, "going forward" and you just said that it did [affect green card holders].
The two would go back and forth for a few more minutes over green cards and the executive order.
Other aides sought to play down the impact of the order.
"There is no evidence that refugees - the most thoroughly vetted of all people entering our nation - are a threat to national security", said Lena Masri, national litigation director for The Council on American-Islamic Relations.
Even after widespread backlash from politicians, pundits, and protesters over the weekend, Priebus said the Trump administration is not going to apologize for the order.
"These are people who traveled overseas, were radicalized, were trained, and then came back and did their bloodletting, their massacre, here on American soil", Conway said. The Boston Marathon bombers were from Kyrgyzstan. "I love you but I mean, let me answer the question", Priebus said.
Kellyanne Conway, counselor to the president, said on "Fox News Sunday" that the seven countries covered by Trump's travel order had been originally flagged by former President Barack Obama's administration. But you bring up a good point and perhaps other countries need to be added, but this is all done for the protection of Americans, and waiting another three days and waiting another three weeks is something that we don't want to get wrong.
PRIEBUS: I'm not suggesting that.
White House spokesman Sean Spicer said Sunday that the controversial ban is a 90-day pause to allow the Trump administration to establish enhanced vetting restrictions.
"These are countries that harbour and train terrorists".
"We're going to have a very, very strict ban and we're going to have extreme vetting, which we should have had in this country for many years".
"It's hopefully going to be decided in the courts as to whether or not this has gone too far", McConnell, a Kentucky Republican, said on the ABC program.
During an interview on "Meet The Press", the NBC anchor asked Reince Priebus to clarify many parts of the executive order, which left airports in chaos Saturday as travelers from seven countries singled out in the measure were detained or put on return flights by border and customs agents.