The lawmakers who failed Thursday to win a key vote on the immigration reform bill before the Senate said on Friday that they will continue to push the bill forward and believed they could still find a compromise that would pass.

"We are not giving up. We are not giving in," Sen. Edward Kennedy, D-Mass., told reporters. He is the chief Democrat at the negotiating table for the immigration bill.

"When it is recognized by the American people that the Senate has not acted (on immigration), I believe there is going to be a wave of support for what we have been trying to do," said Sen. Arlen Specter, R-Pa., who also is on the negotiating team.

The bill, which had been widely criticized, died on a procedural motion Thursday night. But Republican Sens. Specter, Jon Kyl, Lindsey Graham and Mel Martinez were all upbeat after a vote to end debate failed 45-50, failing to reach the 60-vote threshold to move toward final passage.

Click here to see how your senator voted on ending the immigration debate.

Despite the fact that it was primarily Republicans who voted against the maneuver, all the GOP lawmakers who spoke with FOX News were upbeat that the legislation could be revived soon — even within a matter of weeks, with one negotiator noting that last year's bill was first pulled from the floor by then-Majority Leader Bill Frist before it was brought back up again and passed.

Graham said he talked extensively with Senate Majority Leader Harry Reid and is confident the bill will return for senators to take another crack at developing a comprehensive plan to legalize millions of foreigners living unlawfully in the United States.

"I know where the votes are for final passage. ... We're going to get this done," Graham said, adding that the topic is not going to go away. "All I can say is, if you name a post office, you're going to be talking about immigration."

"There are ways we can do this," Reid said later. "There can be an agreement on the number of amendments. Hopefully we can do that in the next several weeks. We're very close."

Kyl, the chief Republican negotiator, told reporters on Friday that he believed a list of about 10 amendments would satisfy the concerns primarily on his side of the aisle, and that those could be considered in no more than three days on the Senate floor.

Kyl, of Arizona, called on Senate leaders to give the time to the bill -- which in the short-term might seem large, but in the long-run would seem small.

"Who will care whether it was an extra three days if we can achieve the result that we're talking about?" Kyl asked.

Reid said support for the bill exists across the country despite repeated polls showing growing opposition to the thrust of the bill and many of its specifics — particularly on the issues of legalization for illegals here now, a guest worker program and chain migration.

"There are a lot of good things in this bill," Reid said. "I'm a creature of the Senate. I understand we live by the rules that govern this body. I accept that. We're going to do everything we can to pass this bill as soon as we can. When is that? I don't know, but we're going to work hard and try to put aside the hurt feelings that we have. The country needs and the Senate needs to do this."

In the meantime, Reid said, President Bush, who champions the comprehensive reform bill, needs to work harder to get Republicans in line.

"Where are the president's men?" Reid asked. "Where are the president's people. I want to finish this bill. But I can't do it alone, we (the Democrats) can't do it alone. We need some help. And I would hope the president understands....he has a relatively short period of time to help us with this piece of legislation."

White House counselor Dan Bartlett told FOX News that he believed there was still hope for the bill to be revived.

"I think its premature to declare this legislation dead. I know the leadership is still talking. The president urges Sen. Reid to reconsider and work with both Republicans and Democrats and get this bill back on track," Bartlett said.

"Now it's time for every member of Congress, particularly in the Senate, to redouble their efforts. The president will following this very closely, as he continues the discussions here in the G-8 [conference in Germany]. There's no more important issue facing the American people back at home," he said.

The legislation up for debate included a temporary guest worker program and a pathway to legalizing the estimated 12 million or more illegal immigrants in the U.S. It also offers provisions to tighten borders and institute a new system to prevent employers from hiring undocumented workers.

As the Senate drew closer to a vote on the bill, Commerce Secretary Carlos Gutierrez and Homeland Security Secretary Michael Chertoff lobbied lawmakers to pass the bill, spending nearly three hours in negotiations in a back room off the Senate floor.

Upon leaving, both men shook bill sponsor Kyl's hand and showered him with praise and encouragement, saying the bill would get done. Gutierrez promised, "We're going to get there. No problem."

Senate Minority Leader Mitch McConnell agreed that the Senate was close to passage of this bill. McConnell, R-Ky., said the bill failed because Democrats tried to rush it, not allowing Republicans to offer key amendments — like tougher border security measures and legalization process for illegal immigrants — that could win over opponents. All but seven Republicans voted against ending the debate.

"Both of us desire the same result, which is to get a bipartisan immigration bill that would be an improvement over the disastrous status quo we have now," McConnell said of himself and Reid.

"I think we were very close to getting there," McConnell added. "We could have finished this bill in a couple of more days in my judgment. We're giving up on this bill too soon. I think we are within a few days of getting to the end of what many would applaud as an important bipartisan accomplishment of this Congress."

Eleven Democrats also voted against ending debate, even though many who supported it had complained that the bill created a class warfare scenario that locks temporary workers into second-class citizen status and rips apart families by favoring employability over blood ties in the approval of future immigrants.

Sen. Edward Kennedy, D-Mass., called the death of the bill, "a disappointment" but held out hope the measure would survive.

"This issue isn't going away," Kennedy said. "Doing nothing is not an option."

Specter, one of the negotiators in what had been dubbed the "grand bargain," condemned the death of the bill, arguing there was opposition on both sides of the aisle but that Republicans did more to hopelessly stall the bill than did Democrats.

"The Democrats were wrong but the Republicans were wronger, to use a word that doesn't exist."

Specter also said the Senate was diminished by the debate and its inability to pass the bipartisan compromise.

"To listen to the debate the last several days, I think people wonder just what is going on," Specter said. "We crafted a bill, bipartisan ... and as of this moment we have not succeeded. I believe we will yet succeed. Accusations have been made that it is amnesty but the fact is if we do nothing we have silent amnesty. This matter is on life-support but it is not dead. It is not moribund."

While Reid insists the bill is not dead, a crowded Senate calendar complicates its prospects. Reid immediately moved onto energy policy after the vote.

FOX News' Major Garrett and Trish Turner contributed to this report.