A day after labor and business groups struck a deal on a new guest-worker program, key senators said they are within reach of finalizing an immigration overhaul proposal.

Members of the Senate’s bipartisan “gang of eight,” which is crafting a comprehensive immigration bill, said they have not signed off on a proposal just yet, but are likely to unveil it next week.

“With the agreement between business and labor, every major policy issue has been resolved,” Sen. Charles E. Schumer said Sunday on NBC’s “Meet the Press.”

Sen. Jeff Flake, Arizona Republican and another member of the gang, said they are “much closer” with labor and business coming together on the guest-worker issue.

“That doesn’t mean we’ve crossed every ‘i’ or dotted every ‘t,’” he said. “We’ve still got a ways to go in terms of looking at the language, making sure that it’s everything we thought it would be,” he said. “But we’re closer, certainly.”

Mr. Flake says he has no intentions of walking away from the debate, but he acknowledged that Mr. Rubio, Florida Republican, is an influential voice in the debate.

“He’s extremely important to this effort,” Mr. Flake said.

“Reports that the bipartisan group of eight senators have agreed on a legislative proposal are premature,” said Sen. Marco Rubio, a Florida Republican who is among the lawmakers working on legislation…

“Eight senators from seven states have worked on this bill to serve as a starting point for discussion about fixing our broken immigration system,” Rubio said. “But arriving at a final product will require it to be properly submitted for the American people’s consideration, through the other 92 senators from 43 states that weren’t part of this initial drafting process.”

Fellow Republican, Rep. Peter King of New York, was skeptical about any prospects for a deal.

“Eight guys in a room saying the border is going to be secure is not enough,” said King, who is not working on the bipartisan proposal.

Marco Rubio, the leading Republican on the Senate’s Gang of Eight immigration reform team, has sent a strongly-worded letter to Patrick Leahy, chairman of the Senate Judiciary Committee, warning against an expected Democratic effort to rush an immigration bill through the committee. Specifically, Rubio suggests — in wording that is actually more than a suggestion but less than an outright demand — that Leahy hold multiple hearings on the Gang of Eight proposal, once it is actually written and introduced in the committee.

“I am aware that the Judiciary Committee, both under your leadership and under the leadership of your predecessors, has conducted a number of hearings related to immigration reform,” Rubio, who is not a member of the Judiciary Committee, writes. “I am certain that those hearings deepened your knowledge of these issues and will guide much of your work this Congress. But they cannot be a substitute for fresh hearings to consider specific legislation as part of a national conversation.”…

“I cannot urge strongly enough that [discussion of the bill] start with meaningful hearings,” Rubio said. “Of particular importance is a full consideration of border security proposals, including testimony from border security experts, the Secretary of Homeland Security, and others. A key feature of our bipartisan approach has been an insistence on meeting border security and other enforcement triggers before unauthorized immigrants can apply for permanent residence. But the success of these triggers will require examining what the American taxpayer’s commitment must be in order to make this security plan a reality.”…

Rubio’s letter places him firmly in line with the Republicans on the committee — Sessions, Charles Grassley, Orrin Hatch, John Cornyn, Mike Lee, and Ted Cruz — who have publicly called for a more deliberative process than Leahy has suggested.

Axelrod on NBC’s “Meet the Press” dismissed GOP suggestions that President Obama was not serious about immigration reform and sought to use the issue to score political points and win Hispanic voters.

“I understand their paranoia, because it was a terribly different issue for them and continues to be,” said Axelrod. But, he added Obama “wants this accomplishment.”

“This is a legacy item for him. There’s no doubt in my mind that he wants to pass comprehensive immigration reform,” he said.

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“I think we’ve got a deal, and we’ve got to write the legislation,” Graham said on CNN’s “State of The Union.” Graham is part of a bipartisan group of eight senators working an effort to develop broad immigration reform legislation…

He expressed confidence it could pass both the Senate and the GOP-controlled House.

“It’s got to be written up. We haven’t signed off; there’s a few details yet. But conceptually, we have an agreement between business and labor, between ourselves. It has to be drafted; it will be rolled out next week” said Graham.