Speaker John Boehner and House Republicans are hoping to kill two birds with one stone.

Before the Senate voted Tuesday against proceeding on a House-passed Homeland Security Department funding bill, Boehner called on Senate conservative firebrands to help pass the measure, loaded with provisions blocking President Obama's immigration policy. At the same time, he and his lieutenants took aim at vulnerable Senate Democrats, daring them to vote "no."

"It's time for Senator Cruz and Senator Sessions, and Senate Republicans and Senate Democrats to stand together with the American people and block the president's actions," Boehner told reporters Tuesday morning.

Boehner's pleas appear to have fallen on deaf Democratic ears: Only 51 senators -- all Republicans -- voted to invoke cloture Tuesday on the House bill, well short of the needed 60 votes.

The peculiarity of Boehner name-checking Ted Cruz and Jeff Sessions cannot be overlooked. Both have railed against Boehner and his team, particularly on immigration issues, calling on them to bring more conservative bills to the floor. Sessions helped scuttle a border-security bill Boehner hoped to bring to the House floor last week, holding that it did not do enough to secure the border or deal with interior enforcement. Cruz, meanwhile, was instrumental in pushing House Republicans to shut down the government in 2013 over demands that any funding legislation block implementation of Obama's health care law.

But now the tables have been turned. Boehner's subtle jab at the two senators betrays an increasing frustration among House Republicans with GOP senators pushing the House to pass legislation they do not have the votes to pass in the Senate. As Tuesday's vote demonstrated the Senate could not pass the House version of the DHS bill, expect House Republicans to blame Senate Republicans, now in the majority, as much as their Democratic counterparts.

"The fight is now in the Senate. It will be won or lost there," said a source familiar with Boehner's thinking, before Tuesday's vote. "Senators like Ted Cruz and Jeff Sessions need to show the American people how we succeed in the Senate. What's their plan to get 'yes' votes from senators like [Claire] McCaskill and [Joe] Donnelly, who've expressed concerns, and ultimately what's their plan to get this passed in the Senate?"

Boehner trained his ire on McCaskill and Donnelly, too, specifically calling them out for issuing statements blasting Obama's executive action granting legal work status to millions of undocumented immigrants.

"There was a whole host of Democrats who issued press releases criticising the president's executive overreach," he told reporters. "Was it all talk?"

In a private meeting of House Republicans on Tuesday morning, Boehner urged his conference to fight on this issue. His top chairmen emerged from the meeting adamant that they would not consider their next steps until the Senate acts one way or the other on the DHS bill.

"They should pass the bill, which funds a very vital national security agency but also turns back this blanket amnesty which is illegal and unconstitutional," House Appropriations Committee Chairman Harold Rogers said. "We think that the vote to bring the bill up on the Senate floor, that motion to proceed, should pass. Anyone who votes against the motion to proceed to fund the Coast Guard, the Secret Service, and all of the security agencies that is in that bill, that vote is critical, and I think people will pay attention to it."

House Homeland Security Committee Chairman Michael McCaul said he, too, hoped the Senate would pass the House bill, but if not, leadership will have to start gaming out the next steps.

"There will be some intense discussions probably this week about how to respond to that. We need to see what they're going to do, and right now we're calling for them to pass our bill," he said.

Meanwhile, Boehner is still finalizing plans to hold a House vote authorizing a lawsuit against Obama over his executive action. It is unclear when that vote would come, but with the immigration action set to begin taking effect later this month, time is running short.

Rep. John Carter, chairman of the Homeland Security Appropriations Subcommittee, said it is possible the House could ask for an injunction on the executive action, although he noted that the chamber will have a high hurdle to prove standing in the case.

"It will be an interesting challenge for everybody if the injunctive relief is granted," he said. "It is supposed to be a priority of the court."

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