Immigration may be coming to the Senate floor next week, but the House has become the central focus in the struggle to pass a comprehensive bill.

Bipartisan House negotiators announced a deal in principle Wednesday evening, shortly after Senate GOP backers of that chamber’s bill attempted to address House conservatives’ concerns about border security.

Though the House group lost one of its eight members — Raúl R. Labrador, R-Idaho, who complained the deal wasn’t tough enough on blocking health benefits for newly legalized immigrants — the agreement could be good news for supporters of an overhaul who have worried about the piecemeal approach advocated by some House GOP leaders.

However, the loss of Labrador from the House group could hurt efforts to woo conservatives.

Indeed, the larger problem has been getting buy-in from the House GOP rank and file for an overhaul that includes a path to citizenship, as the Senate bill would. In order to achieve that goal, Senate Republicans said it’s clear their bill needs to have tougher border security provisions.

“The House side is controlled by Republicans. If you want a bill to pass for the president to sign, it’s got to have Republican support, and right now, the bill that we have in the Senate needs to be strengthened on the border security elements,” Sen. Jeff Flake, R-Ariz., said as he left a briefing with House conservatives in the Republican Study Committee.

“That’s what we’re working on, and we want a bill that passes,” said Flake, who is a member of the bipartisan “gang of eight” that drafted the measure.

Indeed, GOP members of the Senate gang of eight appear to be pushing their fellow group members to back stricter border control measures to pass a bill. For example, Sen. Marco Rubio, R-Fla., suggested on a conservative radio show Tuesday that he might not vote for his own bill if some conservative-backed border amendments don’t pass on the Senate floor.

On Wednesday, Rubio was among the Senate Republicans who traveled across the Capitol to talk to House members. “The general message that I take away from my conversations with Republicans and people across the country is that they are generally prepared to do immigration reform so long as we can ensure that it doesn’t cost the taxpayer money and so long as we can ensure that there isn’t another wave of illegal immigration in the future,” Rubio said.

The bill that Rubio, Flake and six others authored is headed for the Senate floor next week.

But Rubio and Flake were hardly alone in addressing the RSC. GOP senators who have vigorously opposed the gang of eight’s bill, such as Ted Cruz of Texas, Mike Lee of Utah, Rand Paul of Kentucky and Jeff Sessions of Alabama, also addressed the House GOP group.

Paul said there is a divide between the two chambers over a pathway to citizenship. The Senate bill includes a pathway for unauthorized immigrants who meet specific criteria.

“So, you have about 100 congressmen [in the room] and you got a feel for where they are. They’re definitely in a different place than the Senate,” Paul said.

“I don’t think there was any sentiment in the House for a new pathway to citizenship,” he continued. “That was one of the sort of things that seemed to be pretty adamant from most of them.”

But, in what might be a sign of the complexity of the issue, Flake said he doesn’t believe a majority of House members would oppose such a path.

“There are some who don’t want any immigration reform, no path to citizenship, but that view is not shared by all. In fact, I don’t think it’s shared by most,” Flake said. “People just want to make sure we don’t return to this problem years from now.”

Flake also stressed that the House has to be on board to send a bill to the president. He suggested that boosting border security provisions in the bill could accomplish that.

Indeed, Rubio hinted that his vote depends on that. “I think it needs improvements in the border security section,” Rubio said, reprising comments he made Tuesday night on the program hosted by conservative talk show host Hugh Hewitt.

In that interview, Rubio suggested he could vote against the bill he helped write if it didn’t have stronger border security provisions.

There is also the political reality that House resistance to an immigration overhaul along the lines of the Senate bill makes finding votes among senators tougher. Lawmakers as a rule don’t like to vote for controversial provisions if they aren’t going to become law.

Flake, a former House member, said that the supporters want the House to work its will and hopefully come up with a bipartisan proposal that can pass the chamber. But some influential Republicans, including House Judiciary Chairman Robert W. Goodlatte, R-Va., are leaning toward a more piecemeal approach.

“They are hoping to pass their own bill,” Flake said. “Whether they pass a comprehensive bill or just constituent elements of the bill, we don’t know. Either way we can conference with the House. But it’s better for us if we have a bipartisan bill that looks, I’m not saying more like the Senate bill, but is more comprehensive.”

Meanwhile, in announcing his departure from the House immigration group, Labrador warned that the issue of health subsidies could bring down the bill.

“If they are going to have the benefit of living in the United States, which is a privilege not a right, they should provide their own health insurance,” he told reporters.

“I think you’re going to see in the end, this might be the issue that breaks down immigration reform,” he added.