The Trump administration is standing firm over its ban on immigration from seven countries despite court rulings and mass protests against it.
In a statement, President Trump said visas would once again be issued once "the most secure policies" were in place, and denied it was a Muslim ban.
The move has been widely condemned.
Sixteen state attorneys general have said the order is unconstitutional. Several federal judges have temporarily halted the deportation of visa holders.
Mr Trump's executive order, signed on Friday, halted the entire US refugee programme for 120 days, indefinitely banned Syrian refugees and suspended all nationals from seven Muslim-majority countries.
Those who were already mid-flight were detained on arrival - even if they held valid US visas or other immigration permits. It is not known how many others were turned away at airports overseas as they tried to board flights to the US.
Thousands gathered at airports around the country to protest on Saturday, including lawyers who offered their services for free to those affected.
Further demonstrations were held on Sunday, including protests outside the White House and Trump Tower in New York.
Who is affected by the ban?
As well as the ban on all refugees, travellers who have nationality or dual nationality of Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen are not permitted to enter the US for 90 days, or be issued an immigrant or non-immigrant visa.
This includes those who share dual nationality with allied countries, including the UK, although Canada has been told its dual nationals are not affected.
But there remains much confusion.
The UK Foreign Office put out a statement saying that only those dual nationals travelling from one of the blacklisted seven countries would be subject to extra checks - those travelling between the UK and US would not be affected.
However, one Scottish veterinary student - who travels on an Iranian passport - was
White House Chief of Staff Reince Priebus said US Green Card holders - legal residents - would also not be affected, but some have been detained since the order came into effect.
What does the White House say?
Mr Trump tweeted early on Sunday that the US needed "extreme vetting, NOW" but later, in a statement, tried to offer more reassuring words, saying: "This is not about religion - this is about terror and keeping our country safe.
"We will again be issuing visas to all countries once we are sure we have reviewed and implemented the most secure policies over the next 90 days," he said.
In a tweet on Monday morning, Mr Trump placed the blame for any perceived chaos at the airports on protesters, "the tears of Senator [Chuck] Schumer" and Delta Airlines, which had a computer issue on Sunday which led to the cancellation of 170 flights, and another 110 on Monday.
Delta has more than 5,400 flights a day.
Mr Trump also said "only 109 people out of 325,000" travelling, had been detained for extra questioning - a figure Mr Priebus had given to the media on Sunday.
But the statements have failed to allay concern among some in the Republican party. The Republican chair of the US Senate Foreign Relations committee, Senator Bob Corker, said the executive order had been "poorly implemented", particularly for Green Card holders, and the "administration should make appropriate revisions".
Democratic Senate Minority leader Chuck Schumer said the US now appeared "less humanitarian, less safe, less American" and said the Democrats would introduce legislation to overturn it.