Ben Marter, a spokesman for Mr. Durbin, immediately attacked their assertions.

“Credibility is something that’s built by being consistently honest over time,” Mr. Marter wrote on Twitter. “Senator Durbin has it. Senator Perdue does not. Ask anyone who’s dealt with both.”

The remarks by Mr. Perdue and Mr. Cotton were an escalation from a statement they released on Friday, when they said they “do not recall the president saying these comments specifically.”

They also appear to conflict with the account of Senator Lindsey Graham, a South Carolina Republican who was at the Thursday meeting. Mr. Graham told a fellow South Carolina Republican, Senator Tim Scott, that reports in the news media of Mr. Trump’s language were “basically accurate.” The other lawmakers at the meeting, all Republicans, have either not discussed it publicly or made only vague comments.

The Homeland Security secretary, Kirstjen Nielsen, who also sat in on the meeting, said on “Fox…

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