Rep. Mick Mulvaney (R-S.C.) wants the Senate to change its rules to make it easier to advance a bill to fund the Department of Homeland Security. Not going to happen, Senate Republicans say.

The disagreement emerged Thursday afternoon at a news conference where conservative House and Senate Republicans sought to put up a united front against Senate Democrats and pressure them to end their resistance to a House-passed bill that would fund DHS and undo President Obama's executive actions on immigration.

But even in that setting, apparently designed to show some GOP unity in a week filled with discord between Republicans on opposite ends of the Capitol, not everyone was on the same page.

"If the rules are being used to undermine the Constitution, if the rules are being used to deny the ability to govern, then the rules should be changed," said Mulvaney. "The 60-vote rule is not in the Constitution."

Under Senate rules, 60 votes are needed to advance legislation. Three straight times, Democrats have blocked Republicans by keeping them short of 60 votes on the DHS bill.

Top Democrats are urging Republicans to pass a "clean" Homeland Security funding bill with no immigration restrictions. (Reuters)

Mulvaney suggested forcing senators to wage talking filibusters if they want to block legislation. But the Republican senators at the press conference disagreed.

"I don't think that's an option we are looking at right now," said Sen. Dan Sullivan (R-Alaska).

Sen. Ted Cruz (R-Tex.) agreed, adding: "I think the Senate rules wisely protect the minority."

As majority leader in 2013, Sen. Harry M. Reid (D-Nev.) eliminated filbusters for most executive branch nominations, a maneuver known as the "nuclear option," which drew widespread criticism from Republicans.

Senate Majority Leader Mitch McConnell's spokesman Don Stewart said changing the rules is not an option the Kentucky Republican is considering.

Republicans at the press conference, which was hosted by the Republican Study Committee and the Senate Steering Committee, agreed that Senate Democrats need to allow the DHS bill to move forward and are risking the safety and security of Americans by not doing so.

Cruz called Democrats' actions "reckless and irresponsible."

Senate Democrats have demanded a "clean" DHS funding bill that does not contain provisions designed to undo Obama's immigration actions.

Current DHS funding will expire on Feb. 27.

Other House Republicans have reportedly raised the idea of changing the Senate rules.

"Mitch McConnell can change the rules of the Senate, and this is important enough to change the rules of the Senate," said Rep. Raul Labrador (R-Idaho), National Journal reported.

The Hill reported that Rep. Mo Brooks (R-Ala.) said, "I would submit that it's time for the United States Senate to change their rules, to reflect the will of the American people."