Senate Majority Leader Mitch McConnell (R-Ky.), left, and House Speaker John Boehner (R-Ohio) on Tuesday on Capitol Hill. (J. Scott Applewhite/AP)

The Republican honeymoon is over on Capitol Hill.

Just a month into their taking full control of Congress, Republican leaders in the House and Senate are at odds over how to avoid shutting down the Department of Homeland Security as part of an immigration fight with the Obama administration.

House Speaker John A. Boehner (R-Ohio) and Senate Majority Leader Mitch McConnell (R-Ky.) were at an impasse Wednesday over how to pass legislation to fund DHS before funding runs out on Feb. 27, and tension is high among GOP factions that can agree only that Democrats are to blame.

Now the majority party in both chambers, Republicans are eager to demonstrate that they can govern after a banner election boosted their numbers to historic levels, but with the DHS deadline looming, the party is facing its biggest predicament of the new Congress.

Senate Republican leaders argue that, after three failed attempts, they cannot win approval of a House-passed DHS funding bill that challenges President Obama’s executive actions on immigration, because of Democratic resistance. The House Republican position is that the Senate GOP should keep trying.

During a news conference Wednesday, Speaker John Boehner said that “the House has done it’s job” by passing a DHS funding bill and that it is time for the Senate to act. (AP)

“It’s time for the Senate to do their work,” Boehner said Wednesday after a meeting at the Capitol with rank-and-file Republicans. “You know, in the gift shop out here, they’ve got these little booklets on how a bill becomes a law.”

He added, “Why don’t you go ask the Senate Democrats when they are going to get off their ass and do something other than voting no?”

Boehner’s comments seemed to be a direct response to McConnell‘s statement Tuesday that it was “clear we can’t go forward in the Senate” with the current DHS bill.

Republicans on Wednesday held their most direct talks yet about how to end the impasse but could not agree on a strategy for doing so. A pair of new senators who were in the House last year went across the Capitol to address House Republicans at the morning meeting. Later, a top House Republican spoke at a Senate GOP lunch.

There were no breakthroughs.

“They explained to us how the Senate process works, and we were glad to have some of our former colleagues do that,” Rep. Bradley Byrne (R-Ala.) said of the visit by Sens. Cory Gardner (R-Colo.) and Shelley Moore Capito (R-W.Va.). But he added, “From this House member’s perspective, and I think that I reflect the vast majority of the members of our conference, the Senate needs to do its job. Period.”

Democrats, united in the opposition to any bill that would tamper with the president’s immigration orders, have been relishing the GOP’s problems.

“They’re learning how difficult it is to govern,” said Sen. Charles E. Schumer (D-N.Y.).

House Majority Whip Steve Scalise’s trip to the Senate didn’t appear to resolve problems, either. Exiting the meeting, Scalise (R-La.) called on Senate Democrats to allow the bill to move forward and voice their disagreements by offering amendments, not blocking the legislation altogether.

Senate Republicans are increasingly convinced that the House must come up with a different, more moderate bill that some Democrats can support.

“Clearly the DHS bill, as constructed, is not going to get 60 votes” said Sen. Lindsey O. Graham (R-S.C.). “So we would urge the House to do something new.”

Speaking after the lunch with Scalise, Sen. John Cornyn (R-Tex.), McConnell’s top lieutenant, said, “Both of us have our own math challenges.”

The House-passed spending plan would repeal most of Obama’s immigration decisions, but Senate Democrats have blocked it three times in recent days, leaving Republicans struggling to keep DHS open and avoid being blamed for forcing another partial government shutdown.

Republicans, bowing to conservative pressure, had insisted late last year on funding the department only through the end of this month — hoping to gain leverage to counter Obama’s use of executive authority to curtail deportations for many undocumented immigrants and make other changes to immigration policy.

Senate Democrats are demanding a “clean” bill that does not attack Obama’s immigration moves. In the House, a pair of Democrats unveiled such a bill Wednesday, though Republican leaders are not expected to take it up.

“The administration believes firmly that it is pretty irresponsible for Republicans in Congress to be playing politics with the budget of the Department of Homeland Security,” said White House spokesman Josh Earnest, adding that Republicans had “painted themselves into a corner.”

For years, the GOP turmoil over how to handle their internal political and legislative differences has been confined to the House. But it is now spilling out over to the Senate.

The House critics have some allies in the Senate.

Sen. Ted Cruz (R-Tex.) has faulted McConnell, saying the leader has not been forceful enough with Democrats. In order to extract concessions from Obama, Cruz has suggested using threats to block Obama administration nominees or legislative tactics that would essentially shut down the federal government.

He reminded reporters this week that he opposed setting DHS on a different spending schedule from the rest of the federal government, because doing so gave away “virtually all of our leverage by already funding almost the entirety of the federal government.”

Cruz’s approach has infuriated other GOP lawmakers and aides, who complain privately that the Texas Republican and potential 2016 presidential candidate should be using his growing political profile to attack the Democratic opposition rather than the GOP leadership. One senior GOP aide, who asked for anonymity to speak frankly on the matter, credited other conservatives, including Sen. Jeff Sessions (R-Ala.) and outside tea-party-aligned groups, for calling attention to Democratic opposition.

Democrats, meanwhile, are holding Republicans responsible for the stalemate just as they did during the 2013 government shutdown, which did serious political damage to the GOP.

“I think the Republicans realize that this is a losing strategy,” said Sen. Richard J. Durbin (Ill.), the second-ranking Senate Democrat. “Failing to fund the Department of Homeland Security is not a solid message for the new Republican majority.”

In a sign that could be ominous for the prospect of reaching a deal by Feb. 27, some senior Republicans already began efforts to thrust blame onto Democrats if there is any halt to funding for the security agency, arguing that the Senate filibuster should be viewed as the last act that causes the shutdown.

“I always thought it was the people who were filibustering who were the ones who were going to get blamed. In this case, it isn’t us, nor is it the House,” said Sen. Orrin G. Hatch (R-Utah).

With Congress set to be in recess next week, a sense of urgency is beginning to sink in among rank-and-file members.

“One way or the other, we have to get the funding,” said Rep. Peter T. King (R-N.Y.).

Paul Kane and Steven Mufson contributed to this report.