Several other participants in the meeting told the Times they did not recall the president using those words.
Citing six officials who attended the meeting or were briefed about it, The Times explains Trump was outraged after seeing a document showing the number of visas granted by country during the first five months of his administration.
The president lamented that more than 2,500 recipients were from Afghanistan, expressing concerns about terrorism.
Read the full New York Times story here.
Reports of Trump's remarks come as his administration has sought to dramatically reshape the United States immigration system from almost every angle possible.
'General Kelly, General McMaster, Secretary Tillerson, Secretary Nielsen and all other senior staff actually in the meeting deny these outrageous claims.
Trump Said Haitian Immigrants "All Have AIDS" & It Gets Even Worse
Haiti sent 15,000 people. The 40,000 Nigerians who had obtained visas would never "go back to their huts" in Africa, he fumed, according to the two officials-one who was at the meeting and another who heard about the comments from another person in attendance.
They said that on the day in question, Trump stormed into the Oval Office clutching "sheets of papers" which revealed how many immigrants had been allowed into the United States since he took office.
Secretary of State Rex Tillerson, White House Chief of Staff General John. Tillerson responded by throwing up his arms in frustration and arguing that if he was so bad at his job, maybe he should stop issuing visas altogether, according to The Times.
The report comes on the heels of the Ninth Circuit Court of Appeals' ruling Friday that Trump's travel ban, the third he has proposed since becoming president, is illegal.
The White House did not deny the subject of the meeting, but strongly denied that Trump ever used the words "AIDS" or "huts" to describe immigrants entering the U.S. "He's come into office with an aggressive strategy of trying to reverse the demographic changes underway in America".
The Times report said Trump has long held contempt for immigrants, which is likely a surprise to few.