Although Haiti does have one of the highest rates of AIDS in the world at around 2.1 percent of adults, it is dropping.
The month before, when a man who killed eight people in Manhattan was revealed to be originally from Uzbekistan, Trump bemoaned the air of complacency around immigration reform, calling current laws "a joke", the Times reports.
Trump was specifically angered that thousands of foreigners had obtained visas during the first year of his presidency, even though he promised campaign supporters he would curtail immigration.
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.
As the meeting continued, John F. Kelly, then the secretary of homeland security, and Rex W. Tillerson, the secretary of state, tried to interject, explaining that many were short-term travelers making one-time visits.
"They all have AIDS", Trump said about the Haitians, reported The Times, quoting one person, unnamed, who was at the meeting and another who had been briefed by a different person who was in attendance.
It, however, denied that Trump used the words "AIDS" or "huts" in referring to people from any country.
White House press secretary Sarah Sanders issued a statement blasting the paper and denying that Trump had made the comments.
The Times cited six officials who were either there or who were briefed on the meeting in its report.
The Times said its report was the result of more than three dozen meetings.
"General Kelly, General McMaster, Secretary Tillerson, Secretary Nielsen, and all other ranking staff very the gathering deny these absurd cases and it's both tragic and telling the New York Times would print the lies of their unknown "sources" in any case", Sanders said.
Mr Trump has tried to crack down on immigration into the United States, citing terrorism fears, but has been stymied by the courts.
"Two people who described the comments found them so noteworthy that they related them to others at the time, well predating [White House] pushback", Haberman said.