Updated

Hoping to salvage an immigration reform bill that many thought had taken a fatal blow last week, President Bush told Republican lawmakers on Capitol Hill Tuesday that failure to adopt a new bill will leave in place an "unacceptable" status quo.

During a policy lunch, Bush appealed to members of his party — divided over the effort to give residency rights to 12 million illegal immigrants — to revive the bill. The legislation also tightens border security, implements a system to prevent employers from hiring undocumented workers and creates a temporary guest worker program for future migrants.

Some of the lawmakers Bush met with "believe that we need to move a comprehensive bill, some don't. I understand that. This is a highly emotional issue," the president said after the meeting.

"We have got to convince the American people that this bill is the best way to enforce our borders. I believe without this bill, it's going to be harder to enforce the border. The status quo is unacceptable. And I want to thank those senators on both sides of the aisle that understand the time is now to move a comprehensive piece of legislation and the White House will stay engaged," Bush said.

Sen. Trent Lott said lawmakers offered several questions and comments to the president, who was anxious to work with the Senate to get the job done.

"The president made it clear to me and to others that he does not want just any bill, won't sign a bad bill, but he thinks this is an issue that needs to be addressed for the benefit of our country,” Lott, R-Miss., said. “I hope that the majority leader will work with us in a way to get it up in a fair process to move it forward.”

“We’ll move on to immigration when they have their own act together,” Senate Majority Leader Harry Reid said in response.

Nonetheless, several Democrats and Republicans expressed their hope of saving the bill after the Senate voted 45-50 last Thursday to block an end to debate and prevent the Senate from moving to final arguments before a vote on passage.

Opponents said the bill still offers "amnesty" to millions of people who defied U.S. law and crossed the borders without permission. Still, most lawmakers recognize that many of those immigrants have since integrated into the workforce and many are raising their families in the U.S.

They have for the most part dropped calls for these illegals to be removed from the country, though the legislation does require heads of households seeking permanent resident worker status to "touch back" to their home countries to apply for legal work papers.

At the same time, lawmakers also recognize that the status quo cannot be sustained, with an estimated average 54,000 illegal immigrants entering the U.S. each year.

"We think that there's going to be a strong support on final passage when the bill is brought up and finally they've concluded the debate," White House spokesman Tony Snow said before the president's meeting. Snow noted that Republicans killed the bill only after Democratic leaders denied them the opportunity to amend the legislation.

Anticipating Bush's arrival, Democratic Sen. Edward Kennedy said he was hopeful the president, who has made immigration reform his top domestic priority, would help make the difference.

"Like the president, many of us are very determined to get the job done and get it done now. The center is holding together: last night we met to discuss the options for moving it forward and today we'll continue that work. We share the sense of urgency that this important issue deserves," Kennedy said in a statement.

Kennedy's statement follows a letter to Bush penned by Senate Democratic leaders urging him to lean on Republicans to back the measure.

"It will take stronger leadership by you to ensure the opponents of the bill do not block its path forward," the letter said. "Simply put, we need many more than seven Republicans" to support the bill.

But Republican opponents sent their own letter to Bush on Tuesday, urging his administration to follow through with the provisions of last year's immigration reform bill, which called for hundreds of miles of fence to be built along the U.S.-Mexico border.

"We respectfully ask that your administration enforce the border security laws that have already been authorized by Congress regardless of whether the Senate passes the immigration reform bill," the letter from nine Senate Republicans reads.

"The bill assumes that several critical border security benchmarks can be achieved within 18 months. These security triggers are already authorized under current law and can be completed without the immigration bill," the letter continues.

According to a FOX News-Opinion Dynamics poll released last week, a majority of those surveyed — 58 percent — said they want the federal government to enforce existing border security laws. Thirty-four percent said the entire immigration process needs to be scrapped and reworked. But support for allowing illegal immigrants to stay in the United States is 67 percent when conditions of paying taxes and obeying the law are met.

Click here to see the poll results.

Proposed Resolution to the Standoff

One of the proposals to overcome the impasse is to offer a limited list of Republican-sought amendments that could be considered before a final vote on the measure.

Bill supporters say, for now, their strategy is to get a list of proposed amendments from the authors of the border security letter — Sens. Jim DeMint of South Carolina, Tom Coburn and Jim Inhofe of Oklahoma, Mike Enzi of Wyoming, David Vitter of Louisiana, Jim Bunning of Kentucky, Charles Grassley of Iowa, John Ensign of Nevada and Jeff Sessions of Alabama.

“The details of exactly how we get from where we are now to the finish line is not something we’re prepared to announce today but I do think this bill is about 80 or 85 percent of the way through toward the finish line and we don’t have any interest in giving up on it,” Senate Minority Leader Mitch McConnell said after meeting with the president.

Sen. Jon Kyl, R-Ariz., one of the original bill's authors, told FOX News on Tuesday that Republican leaders had received a list of 12 amendments that could resolve the standoff. However, some of those amendments are bill killers. The "grand bargainers," or moderate lawmakers from both parties who developed the legislation, planned to meet at 5 p.m. ET to cull the list to be presented to Reid.

"I think once we've allowed them to have a couple of amendments up, it's pretty hard for them then to argue that they haven't had their chance," he said Monday before the amendment list was formed.

But Sessions told FOX News he won't agree to anything that will get the existing legislation passed. He called the compromise "a cover — just a chance to try to get votes, but it's not going to mean anything."

"They select the amendments, not the senators who are offering them. They've already agreed for the most part to vote against any one of them that has any significant change in the bill, and I believe that the bill needs fundamental change so I don't see any need to agree to that," Sessions said.

Kyl said if opponents are still dissatisfied with the bill after a series of votes, he, McConnell, Lott and as many Republicans as they can find will move to support ending debate and voting on final passage. In other words, they'll roll over the dissenters, in the words of Sen. Arlen Specter, R-Pa.

Kyl said the legislation is the last best hope for getting into place a system that can identify whether immigrants are legally employed or not.

Still unclear is whether Bush will have the influence among Republicans to move the process forward. Reid, D-Nev., said he will bring up the measure again if Democrats can be assured of more Republican backing.

FOX News' Trish Turner and The Associated Press contributed to this report.