The remarks were made after a rally, and there was speculation by some on social media afterwards that Trump did not hear the entirety of the question, because he refers to the border in his answer.
"There should be a lot of systems, beyond database, we should have a lot of systems, and today you can do it," Trump responds. "But right now we have to have a border, we have to have strength, we have to have a wall. And we cannot let what’s happening to this country happen."
"But that's something your White House would like to implement?" the reporter presses.
"Oh I would certainly implement that. Absolutely." Trump answers.
But Trump continues to talk about the logistics of implementing such a database at length.
"But for Muslims specifically, how do you actually get them registered in a database?" the reporter asks. Trump responds: "It would be just good management. What you have to do is good management procedures and we can do that."
He also responds to a question about going to mosques to "sign these people up." "Different places, you sign them up at different, but it’s all about management, our country has no management," Trump says.
Trump tweeted afterwards that he didn’t suggest the idea of a database -- but did not retract his response.
In an interview with Yahoo News in November 2015, Trump did not rule out the idea of a database or a special form of identification for Muslims.
Trump was asked if that would include “registering Muslims in a database or giving them a form of special identification that noted their religion.”
Trump did not say yes or no, instead saying: “We’re going to have to look at a lot of things very closely We’re going to have to look at the mosques. We’re going to have to look very, very carefully.”
The Des Moines register subsequently asked about the Yahoo interview.
The report says the first time Trump was asked about his remarks, he said: "I certainly wouldn’t want to do it but we have to be vigilant, we have to be very careful."
The second time he was asked, according to the Register, Trump responded that he had never heard of such a thing. “I didn’t say that. I never said that,” he said, according to the Register.
ABC News' Katherine Faulders and Alex Mallin contributed reporting.