Trump said the USA should instead welcome more people from countries such as Norway.
Commenting on the White House meeting, Graham said on Friday that diversity had always been the United States' strength.
Durbin, the No. 2 Democrat in the Senate, called the White House to tell Trump that a bipartisan group of senators had struck a deal to put Dreamers on a path to citizenship, beef up border security and take other steps to change immigration laws.
The President has since denied the comments; however, following Trump's denial, Sen.
But Democratic Sen. Dick Durbin, who also attended the meeting, told reporters: "That's what he said".
"He said these hate-filled things and he said them repeatedly", Durbin said.
Graham declined to speak to CBS, but on Friday afternoon also released a statement that said, "Following comments by the President, I said my piece directly to him yesterday".
'I appreciate Senator Durbin's statements and have enjoyed working with him and many others on this important issue.
Richard Blumenthal (D-CT) said Trump's comments "smack of blatant racism - odious and insidious racism masquerading poorly as immigration policy".
There was no mention at that time of the language that sources said Trump had used, details of which were soon leaked.
In a statement it said it had asked the U.S. government, through its ambassador, to "clarify" if the derogatory remark also applied to Botswana given that there were Botswana nationals living in the United States and others who wished to go there. "Made up by Dems", Trump wrote.
The reference to Norway may have been prompted by Prime Minister Erna Solberg who visited the White House on Wednesday when the president praised Norway for running a trade deficit with the United States and for buying US military equipment.
After the Thursday meeting at the White House - and a few hours before The Washington Post reported news of Trump's "shithole" remarks - Graham was calm and composed when speaking with reporters about how his pitch had gone over with the president and his colleagues.
"It makes me embarrassed to have this guy as the president of my country", said Raoul, who's running in Illinois' March primary for state attorney general. They don't blink. They stand tall and they have dignity.
Donald Trump's least presidential moments so far.
"We would not deign to make comments as derogatory about any country facing such difficulties", Duarte said.
Ryan said his ancestors were Irish and 'were really looked down upon.' He called immigration 'a handsome story of America ' and said Africans in his hometown of Janesville, Wisconsin, are 'incredible citizens'.