Many Democrats have said they will not vote for a spending bill this week needed to avoid a partial government shutdown without a deal on DACA.
Trump has not clarified to aides what he said, but told reporters Sunday night in Florida that comments attributed to him "weren't made".
At least one Republican senator has claimed that the story of Trump's controversial remark was a "gross misrepresentation".
The controversy over Trump's language continued after Durbin and other senators questioned Homeland Security Secretary Kirstjen Nielsen - who was also at the meeting - as she testified Tuesday before the Senate Judiciary Committee.
It's weird that people in the room don't remember Trump using that word when Trump himself was calling friends to brag about it afterwards.
It said the "remarks dishonor the celebrated American creed and respect for diversity and human dignity" and that there was a serious need "for dialogue between the US Administration and the African countries". I've always believed that America is an idea, not defined by its people but by its ideals. Members of Congress from both parties are trying to strike a deal Mr Trump would support to extend that protection.
Local leaders say they want the president to apologize. Dick Durbin, who said the president used the term "s-thole countries", referring to him as "Dicky" and blaming him for scuttling a deal to help young immigrants, New York Post reported."Senator Dicky Durbin totally misrepresented what was said at the DACA meeting". As always, I look forward to considering additional ideas that could make the proposal even better.
Mr Trump suffered a backlash over the comments and was accused of being racist, but his son called the suggestions "total nonsense". Let's stop tip-toeing around and making excuses for this president.
What has been the political fallout?
CNN's Don Lemon opened CNN Tonight by saying: "The President of the United States is racist".