Amid a torrent of conservative criticism of the Senate immigration bill, a band of House Republicans is working to bridge the stark divide in their party while trying to get around the “amnesty” label poisonous to the GOP’s base.

Rep. Mike Pence (Ind.), the former Republican Study Committee chairman whose joint immigration proposal with Sen. Kay Bailey Hutchison (R-Texas) went down in flames last year after it was attacked by fellow conservatives, said he sees a growing sense among Republicans for the need to come up with a viable alternative.

“Opposition to the Senate bill among House Republicans is monolithic,” he said, but asserted it is no longer enough to just say no.

“I think there is a greater willingness among Republicans to embrace a proposal we can all be for instead of just being against the Senate proposal, which wasn’t the case last year,” Pence said. He added that he’s working to convince other Republicans to try to build a “no amnesty” bill that would allow illegal immigrants to quickly reapply to come back to the country after they leave.

Sen. John McCain’s (R-Ariz.) challenge to opponents of the Senate bill to propose an alternative deserves a response, Pence said. “I think Sen. McCain’s retort is very fair,” he said. “If you are against the bill, what are you for?”

Pence’s proposal would force every illegal immigrant to leave the country, he said.

“That overcomes the amnesty objection. … Your violation of the law is a function of geography so your correction of that status should also involve geography,” Pence said.

Because Pence’s plan would only require people to leave the country briefly, it has been ripped by many conservatives.

“Mike should have learned his lesson a year ago,” said Rep. Dana Rohrabacher (R-Calif.). The impetus for any immigration legislation isn’t aimed at discouraging illegal immigration, he contended, but rather to facilitate it.

But Pence’s ideas are starting to get traction among some Members who are looking for a “cover” to vote for an immigration bill but are worried about the “amnesty” backlash, said one House Republican supportive of Pence’s ideas.

And Rep. Mike Castle (R-Del.), a leading moderate, said an idea like Pence’s could get some support, but Republicans are waiting to see what the Senate does before making their own proposals. “Any system that would allow somebody who has come here illegally to move ahead of somebody who wanted to come here legally may not be labeled as amnesty, but it is problematic,” he said. Castle added that there is no easy way, however, to deal with the 12 million illegal immigrants already in the country.

“I believe everything hinges on what happens in the Senate,” said moderate Rep. Ray LaHood (R-Ill.), who meets with Democratic Caucus Chairman Rahm Emanuel (Ill.) weekly to discuss strategy.

“I think if the Senate passes a bill that raises the chances of the House passing a bill dramatically. … We have been working very hard to build the votes based on a Senate-passed bill, and we’re making progress,” LaHood said. “All of us who care a lot about this have our fingers crossed.”

GOP Reps. Steve King (Iowa) and Ed Royce (Calif.) held a press conference Wednesday blasting the Senate bill and the Bush administration’s failure to protect the border. Their solution is simply to enforce the laws on the books calling for sanctions against employers and a fence across the Mexican border.

“Why aren’t we simply enforcing the law?” Royce asked.

Grassfire.org, an organization fighting the Senate bill, is launching a new television ad campaign titled “Where’s the Fence?” featuring three old ladies at the border looking for a fence mandated by Congress but not yet built. The ad will air on national cable news networks and in several states.

Like Royce and King, the group is making no bones about targeting fellow Republicans. “Amnesty cannot pass without Republican support,” noted Grassfire.org President Steve Elliott.

Meanwhile, Senate Democrats and Republicans softened their positions considerably Wednesday, stepping back from the strident rhetoric of Tuesday that appeared to put the immigration bill on a path to defeat.

“Everybody was overreacting and jumping the gun yesterday,” Senate Minority Whip Trent Lott (R-Miss.) said Wednesday. “People are more reasonable today.”

Members of both parties expressed optimism that a bill would pass the chamber either by the end of this week or early next week, with Republicans predicting that they would need to go into next week and Democrats continuing to aim for completion by Friday.

Though Senate Majority Leader Harry Reid’s (D-Nev.) decision to file a motion to limit debate, or invoke cloture, on the compromise measure caused much of the partisan furor on Tuesday, it also jump-started negotiations between the parties. Bipartisan negotiators spent the bulk of Wednesday trying to whittle down the list of amendments to be voted on, and Democratic leaders relented on allowing votes on several GOP amendments.

Negotiators largely stuck together in defeating amendments that they say would have gutted their compromise, such as a Sen. John Cornyn (R-Texas) amendment that would have barred felons, including those who have committed immigration offenses, from being eligible for certain visas.

“It suggests there are a lot of people who are willing to swallow hard to get this thing done,” Sen. John Thune (R-S.D.), who is expected to oppose the overall measure, said of Republicans who opposed the Cornyn amendment.

Still, it was unclear whether Republican leaders and GOP members of the “grand bargainers” who crafted the bill would be satisfied with the pace of votes on amendments and whether they would withdraw their threat to vote against cloture — a move that Reid warned Tuesday would kill the bill.

As of press time, Reid still was holding out the possibility of postponing the cloture vote scheduled for this morning to allow more votes on amendments.

Despite winning votes on several of their amendments Wednesday, GOP conservatives showed no signs of supporting an end to debate, and a Senate Democratic source indicated that leaders “are not sure you have the left flank of the party voting for cloture.”

The source added that Wednesday’s developments did not guarantee success for the bill, but “what you’ve really seen today is a renewed commitment to try to figure out a way forward.”

Even if progress on the Senate measure continues to stall, House Majority Leader Steny Hoyer (D-Md.) asserted Wednesday that his chamber intends to move forward with its own bill.

“While the Senate was expected to go first … it was not a condition for us to go forward,” Hoyer said, adding that a markup in his chamber likely will take place later this month, then move to the House floor in July.

But the House could reconsider its options in the event the Senate deadlocks over an immigration package. “We’ll have to see if that happens. … We are still of a mindset to go forward,” Hoyer said.

House Republican Conference Chairman Adam Putnam (Fla.), who generally supports the idea of passing a comprehensive bill, said he feared that even if the Senate passes a measure acceptable to the House GOP, House Democrats would alter it.

“The Senate bill needs to move right,” Putnam said. “The Senate bill has been inching rightward slightly with amendments, but my fear is that all of that will backslide when it comes to the House and it will become even more unpalatable.”

Jennifer Yachnin contributed to this report.