Besides the 120-day ban for refugees, the order also bans entry to the United States for citizens of seven majority-Muslim countries and indefinitely bars travel by Syrians to the US.
At a news briefing with Mr McAleenan, Homeland Security Secretary John Kelly said the travel ban for the seven countries may be extended and other countries could also be added to the list.
“I would be less than honest if I told you that some of those countries that are currently on the list may not be taken off the list anytime soon,” Mr Kelly said.
“They’re countries that are in various states of collapse” and may not be able to verify that people applying to come to the United States are who they say they are, he added.
In his first briefing with reporters since he was confirmed, Mr Kelly defended Mr Trump’s order and said its intention is to keep would-be terrorists out of the United States and not serve as a ban on Muslims.
Early in his campaign, Mr Trump had called for a ban on Muslims entering the United States.
Mr Kelly said “the vast majority of the 1.7 billion Muslims that live on this planet, the vast majority of them have, all other things being equal, have access to the United States”.
The retired Marine general also said the order was “not a travel ban” but a “temporary pause that allows us to better review the existing refugee and visa-vetting system”.
White House spokesman Sean Spicer made that point too in his daily briefing to reporters.
However, Mr Trump referred to it as a “ban” in a tweet on Tuesday defending the decision not to provide advanced notice to travellers. Mr Spicer also called it a ban on Monday.
“If the ban were announced with a one-week notice, the ‘bad’ would rush into our country during that week,” Mr Trump wrote. “A lot of bad ‘dudes’ out there!”
The roll-out of the order has been widely criticised, a point Mr McAlleenan conceded, saying communication among government agencies had “not been the best”.
However, he said the refugees’ waivers were being done in concert with the State Department.
He said the refugees would be processed through the end of the week.
According to guidance provided to some refugee aid agencies by the State Department, none of the refugees are from the seven countries singled out for an all-out travel ban.
Those countries are Syria, Iran, Iraq, Somalia, Sudan, Yemen and Libya.
Mr Kelly denied reports that he had been out of the loop in the White House planning for the immigration restrictions.
He told reporters he looked at two drafts of the order before the Friday signing and that high-level government lawyers and agency officials were involved in drafting it.
He did not provide names of who was involved in the initial planning, but said the group included people from Mr Trump’s transition team.
Mr Kelly also said he knew the order was coming because Mr Trump had long talked about it as a candidate.
People who know Mr Kelly, however, told The Associated Press that he was not aware of the details in the directive until around the time that Mr Trump signed it.