"If a person wants to be the nominee of the Republican Party, there can be no evasion and no games. They must reject any group or cause that is built on bigotry. This party does not prey on people's prejudices," House Speaker Paul Ryan told reporters on Capitol Hill.
Senate Majority Leader Mitch McConnell said plainly, "Senate Republicans condemn David Duke, the KKK, and his racism."
"There has been a lot of talk in the last 24 hours about one of our presidential candidates and his seeming ambivalence about David Duke and the KKK, so let me make it perfectly clear," he said. "That is not the view of Republicans who have been elected to the United States Senate, and I condemn his views in the most forceful way."
Congressional Democrats also seized on the issue Tuesday to argue that Trump's views represented his party as a whole.
"The leading Republican presidential candidate's refusal to disavow the KKK was a breathtaking low-point for our country," House Minority Leader Nancy Pelosi said in a statement. "Yet while Donald Trump's radical agenda does not reflect the values of the American people, it is a perfect reflection of many in the House Republican Conference."
In an interview with CNN's Jake Tapper on Sunday
, Trump declined to disavow support from David Duke, the former grand wizard of the Ku Klux Klan, claiming he didn't know anything about the group.
He later complained he didn't hear the questions in the interview because of a "bad earpiece."
Ryan asserted the Republican Party is "the party of Lincoln."
"We believe all people are created equal in the eyes of God and our government. This is fundamental and if someone wants to be our nominee they must understand this," he said.
Despite his criticism of the current GOP front-runner's recent comments, the speaker repeated his pledge to back whoever secures the delegates to win the nomination.
"My plan is to support the nominee," Ryan said.
As House speaker, Ryan will preside over the GOP convention in July.
Citing his role at the convention, Ryan regularly refuses to weigh in on the presidential campaign, even in the face of a string of controversial statements from Trump.
Ryan did denounce Trump's proposal in December to ban all Muslims from entering the United States, though, and said on Tuesday, "I hope this is the last time I need to speak out on this race."
Pressed about divisions inside his party and warnings about the impact of Trump at the top of the ticket, the speaker downplayed any splits.
"We're conservatives. We believe in the American idea. We believe in the principles that built this country," he said, adding that he would continue working on an agenda for the eventual nominee and speak out nationally about Republicans policies and contrast them with Democratic proposals.