Mr Trump's alleged comments came after he was told 15,000 Haitians had received visas to enter the U.S. since he became President.
In an Oval Office meeting in July, the sources said, Trump read from a list in his hand during a discussion about immigrants who had been granted to enter the country.
According to The Times' report, Trump was upset about the number of people coming into the United States, and said that was "making a mockery of his pledge" to crack down on illegal immigration.
The president complained about the more than 2,500 who came from Afghanistan, which the president reportedly described as a terrorist haven.
Forty thousand had come from Nigeria, Mr. Trump added.
Of the 40,000 Nigerian immigrants who had been granted entry, he is alleged to have said they would "never go back to their huts in Africa" now that they had seen the United States. Kelly (who was at that time the secretary of homeland security), and domestic policy adviser Stephen Miller were all there.
The White House unequivocally pushed back on a report that President Donald Trump talked about workers in a pretentious and belittling design amid a June meeting with top organization authorities.
White House spokeswoman Sarah Huckabee Sanders denied the reports.
According the sources, when Trump learned 15,000 people from Haiti had been granted U.S. visas, he remarked: "They all have AIDS". The explosive and disparaging remarks about immigrants attributed to the president were sourced to a pair of unnamed officials, one who the Times said was present in the meeting, and another who was briefed about the comments by a second attendee. In a shocking story Saturday, The New York Times cited two sources who described in explicit terms how Trump has raged against his advisers for not doing more to translate his nativist politics into policy.
The Times report said Trump has long held contempt for immigrants, which is likely a surprise to few.
"Kelly, General (H.R.) McMaster, Secretary (Rex) Tillerson, Secretary (Kirstjen) Nielsen, and all other senior staff actually in the meeting deny these outrageous claims and it's both sad and telling The New York Times would print the lies of their anonymous sources anyway", Sanders told CBS News.