Donald Trump has suggested terrorist attacks in France and Germany were their "own fault" and citizens of those "compromised" countries could be subject to "extreme vetting" before entering the United States.
And he plans to extend his list of places that could earn travellers extra scrutiny over the coming weeks.
"I'm going to be coming out over the next few weeks with a number of the places."
Mr Trump was giving more detail about a policy he announced in his Republican convention speech on Friday.
When asked in an interview with NBC News if that might lead to a point when not a lot of people from overseas are allowed in, the Republican presidential nominee said: "Maybe we get to that point."
He said France and Germany had "totally" been compromised by terrorism.
Mr Trump said: "You know why? It's their own fault because they allowed people to come into their territory.
"We have to have tough, we're going to have, tough standards. If a person can't prove what they have to be able to prove they're not coming into this country."
He added: "That's why Brexit happened OK? Because the UK is saying 'We're tired of this stuff, what's going on? We're tired of it.'
"Here's my plan, here is what I want, extreme vetting."
Mr Trump also rejected suggestions that his comments on Nato members being required to pay their share in order to get the protection afforded by the treaty were a mistake.
He also disputed descriptions of his convention speech earlier in the week as "dark".
Mr Trump said: '"It was an optimistic speech. I talk about the problems but we're going to solve the problems."
And, the Washington Post reports that Trump made clear in the NBC interview that he has not rolled back his proposal to ban Muslims from entering the United States, despite his top allies insisting that he had.
In accepting the Republican nomination on Friday, Trump said the country "must immediately suspend immigration from any nation that has been compromised by terrorism until such time it's proven that vetting mechanisms have been put in place." Trump made no mention of Muslims in the speech, leading many to conclude that he had formally changed his position - just as a number of his top allies, including his running mate, said he had.
During the NBC "Meet the Press" interview aired yesterday host Chuck Todd asked Trump whether his comment should be interpreted as a "slight rollback".
"I don't think so. I actually don't think it's a rollback. In fact, you could say it's an expansion," Trump said.
"I'm looking now at territory. People were so upset when I used the word 'Muslim': 'Oh, you can't use the word "Muslim." ' Remember this. And I'm okay with that, because I'm talking territory instead of Muslim."
Trump first proposed banning nearly all Muslims overseas from the US in early December, soon after the attack in San Bernardino, California.
Trump's original statement - which calls for "a total and complete shutdown of Muslims entering the United States" - is still on his campaign website. This position continues to be one of Trump's most controversial, and a key reason that some fellow Republicans do not want to help him with his campaign.
After he became the presumptive nominee, Trump made comments that seemed to indicate that he was willing to soften his position.
In May, Trump said the Muslim ban is "just a suggestion" and that he's open to other ideas. In June, after the attack at a gay nightclub in Orlando, Trump called for a temporary ban on "certain people coming from certain horrible - where you have tremendous terrorism in the world, you know what those places are."
At the time, it appeared that Trump was expanding his ban to include more people, not limiting its scope, but his staff would not confirm where their boss stood.
Later that month, during a visit to one of his golf courses in Scotland, a reporter asked Trump whether he would be okay with a Muslim from Scotland coming into the United States, and he said it "wouldn't bother me."
Afterward, spokeswoman Hope Hicks said in an email that Trump's ban would apply only to Muslims in states with high risks of terrorism, but she would not confirm that the ban would not apply to non-Muslims from those countries or to Muslims living in peaceful countries.
More recently, several of his top allies have said that the nominee no longer wants a religion-based ban.
Indiana Governor Mike Pence, Trump's running mate, criticised Trump's original proposal but says he can support Trump's current position, which he described as temporarily suspending "immigration from countries where terrorist influence and impact represents a threat to the United States."
Trump's campaign chairman, Paul Manafort, also has pushed that characterisation of the ban, while dodging questions on whether it would still target followers of Islam.
Reince Priebus, the Republican Party chairman, said last weekend that Trump has moved away from his original proposal and that "there is no religious test on the table."
In the NBC interview, which was filmed on Sunday, Trump said his proposed ban is constitutional.
"Just remember this: Our Constitution is great, but it doesn't necessarily give us the right to commit suicide, okay?" he said. "Now, we have a religious, you know, everybody wants to be protected. And that's great. And that's the wonderful part of our Constitution. I view it differently. Why are we committing suicide? Why are we doing it? But you know what? I live with our Constitution. I love our Constitution. I cherish our Constitution. We're making it territorial. We have nations and we'll come out, I'm going to be coming out over the next few weeks with a number of the places."