The primarily white cover, for the January 22 issue of the magazine, depicts a glimpse of Trump's recognizable hairdo inside a large, black hole.
But Democratic Senator Dick Durbin, who was present at the meeting, told MSNBC on Friday that Trump had used the slur to describe those countries.
The reported language was the latest in a long string of anti-immigrant and anti-Muslim comments by Trump that have been condemned as racist.
Many Africans reminded the USA of its historic role in the continent's woes.
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In a tweet on Friday, Trump denied using the word "shithole", but Durbin insisted that the president had indeed said it.
After making the "shithole" comment, Trump suggested the United States should instead welcome immigrants from places like Norway, whose prime minister he met on Wednesday.
The ease with which he routinely and reflexively denies the truth has been well documented, and many of us continue to marvel at how little Trump seems to suffer the consequences of his dishonesty. Many have children who were born in the States and have U.S. citizenship. I could tell you how I have spent an inordinate amount of time and energy, throughout my life, educating people about Haiti and disabusing them of the damaging, incorrect notions they have about the country of my parents' birth. Alexander Hamilton said that the Louisiana Purchase would have never happened were it not for the "courage and obstinate resistance of the black inhabitants" of Haiti.
"(Trump) is sending a coded message that he wants to return to the good old days, of slavery, the good old days of water hoses, the good old days of German Shepherds biting people fighting for their basic rights", Brooklyn Borough President Eric Adams said.
Several university presidents declined, through representatives, to weigh in on Trump's remarks.
Trump on Friday denied describing certain nations as "shithole countries" after triggering worldwide outrage. Saying they believe the administration has a "huge misunderstanding of the African continent", the union said "there is a serious need for dialogue" between the Trump administration and African countries.
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"The Congressional Hispanic Caucus calls on every member of Congress to reject this vile statement, which erodes the moral fiber that binds us together as Americans". "I'm more troubled by the silence from people who are supposed to be men and women of righteousness".
Emigration from Norway to the US hit its peak in 1882 when nearly 29,000 mostly poor Norwegians crossed the Atlantic. "I've not read one of them that's inaccurate".
In a statement, the Haitian government expressed its shock and outrage. "It is more about ethnicity than shared values", said Hilde Restad, an associate professor in worldwide affairs and a former US resident.
In 1996, Madeleine Albright, who was then the US Ambassador to the United Nations, said that the price of the American attempts to "punish" Saddam Hussein - the death of half a million children due to US-led sanctions against Iraq - was "worth it".
Trump's comments hit hard in Haiti, which on Friday marked the anniversary of a 2010 natural disaster that killed between 220,000 and 300,000 people. We are entrepreneurs, big and small, dreamers, workers.
The Hill reported the words "Pay Trump bribes here" "emoluments welcome", and "we are all responsible to stand up and end white supremacy" were also projected onto the building. We pay more for less.
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Marquise Brown ended with eight catches for 114 yards. "I didn't make the throws I normally make, and it showed", he said. Georgia blocked Oklahoma's attempt at a field goal and only needed a field goal to win it.
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