Sen. Ted Cruz (R-TX) speak to reporters on Dec. 13, 2014 on Capitol Hill in Washington.
Lauren Victoria Burke—AP
February 12, 2015 3:59 PM EST

A vocal crew of frustrated House conservatives called for a change of Senate rules Thursday, sparking a war of words with their Republican Senate colleagues, who defended the rights of Democrats to block legislation to undo legal residency President Obama has promised for millions of undocumented immigrants.

“I think when you have a president doing things that are unconstitutional, everything should be considered to stop that,” says Ohio Rep. Jim Jordan, a conservative leader.

The response was quick from the other side of the Capitol building. “Change the Senate rules?” said Republican South Carolina Sen. Lindsey Graham, who is exploring a campaign for the Republican nomination in 2016. “No way. That’s crazy.”

Even Texas Sen. Ted Cruz and Utah Sen. Mike Lee, two of the most conservative members of the Senate, said they did not agree with their House conservative allies. “For many decades the Senate rules have protected the rights of the minority and as a result the Senate functions, as the Framers put it, ‘as a saucer to cool the hot tempers of the moment,'” Cruz told TIME in a short interview. “The problem is not the Senate rules. The problem is that Senate Democrats have chosen to filibuster funding for the Department of Homeland Security. Democrats right now are engaged in partisan obstructionism.”

At issue is a bill to fund the Department of Homeland Security, which is facing a February 27 funding deadline. If Congress doesn’t pass a DHS spending bill, the agency will partially shut down and potentially furlough some 30,000 employees.

Republicans believe that the deadline gives them leverage to pass riders stripping away the President’s immigration actions, including a 2012 program granting temporary deportation relief to young adults who came to the country illegally as children. But they’ve ran into a united Senate Democratic bloc and failed three times last week to put the House bill on the Senate floor with the 60 votes needed. With only 54 Senate Republicans—and a few publicly calling for a shift in strategy—House and Senate conservatives have split over what to do next.

“They can change them today,” says Idaho Rep. Raul Labrador of the Senate’s filibuster rules. “They don’t need to wait for six or seven senators to decide to bring the bill to the floor.” He referenced the Democratic decision to scrap filibusters of federal judges and other Presidential nominees in the last Congress. “The Democrats changed their rules last time,” Labrador continued. “I know there’s a lot of tricks in the Senate and they should be applying one or all of those.”

Senate Majority Leader Mitch McConnell’s office says no proposal to change Senate filibuster rules is under consideration.

Democrats have called for a “clean” bill without any of the immigration amendments.

“All they got to do is send over a clean DHS bill and it’ll pass in a matter of hours,” says Vermont Democratic Sen. Patrick Leahy.

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