A bipartisan deal on immigration legislation would need tough enforcement and even stricter penalties for those who came to the United States illegally, a leading Republican at the center of negotiations said Sunday.

Sen. Marco Rubio, who’s among the eight senators writing a plan that’s expected to come out Tuesday, tried to promote and defend the framework for the emerging overhaul that would provide a path toward citizenship for those who came to the country illegally or overstayed their visit.

While the deal does include a long and difficult process for the 11 million individuals in question, Rubio insisted the proposal does not include an “amnesty” provision that fellow conservatives have called a deal-breaker.

“We’re not awarding anybody anything. All we’re doing is giving people the opportunity to eventually earn access to our new, improved and modernized legal immigration system,” said Rubio, a Florida Republican and Cuban-American.

But among some of his fellow Republicans, there are serious doubts.

“I’m not convinced,” said Sen. Jeff Sessions, R-Ala. “I know Sen. Rubio’s heart is exactly right. And I really respect the work of the ‘Gang of Eight.’ But they have produced legislation … that will give amnesty now, legalize everyone that’s here effectively today and then there’s a promise of enforcement in the future.”

Sen. Mike Lee, a Utah Republican, added: “The pathway to citizenship, right now, before those other elements are in place, is the deal-breaker for me.”

He said he could consider supporting the proposals only if the first priority were border security.

Rubio said he would abandon the overhaul effort if enforcement, border security and other elements are softened to his dissatisfaction. Rubio also told those immigrants that it would perhaps be easier if they returned to their home countries and started the process from scratch rather than use the process Rubio is proposing.

“So I would argue that the existing law is actually more lenient, that going back and waiting 10 years is going to be cheaper and faster that going through this process that we are outlining,” he said.

Other lawmakers helping to write the legislation acknowledged the political challenges of the issue.

“A lot of my conservative colleagues have significant questions and they’re legitimate,” said Sen. John McCain, R-Ariz. “This is the start of a process, this is a vehicle that requires hearings, requires input and we welcome all of that. … I am guardedly optimistic that we will see finally the end of this long, long trek that a lot of us have been on for many years.”

The measure would put millions who are in the U.S. illegally on a 13-year path to citizenship, while toughening border security requirements, mandating that all employers check the legal status of workers, and allowing tens of thousands of new high- and low-skilled workers into the country with new visa programs.

The legislation is expected to include a new emphasis on merit-based immigration over family ties.

“This is a very balanced bill. The American people have told us to do two things: one, prevent future flows of illegal immigration; and then, come up with a common-sense solution for legal immigration. And that’s what our bill does,” said Sen. Chuck Schumer, D-N.Y.

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All of this, however, is contingent on the border security and enforcement, Rubio said.

“If you are undocumented here now, if you are illegally in the U.S., that you can’t even apply for this until these plans are in place and they begin to implement them,” Rubio said. “And then you’re going to have to pay a fine. You’re going to have to pay an application fee. You’re going to have to pass a background check.”

Without those pieces, the path to citizenship is unavailable, and the proposal is available only for those who arrived in the United States before Dec. 31, 2011. Anyone who came after that date would be subject to deportation.

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Rubio pressed his case during interviews on ABC’s “This Week,” CBS’ “Face the Nation,” CNN’s “State of the Union”, “Fox News Sunday” and NBC’s “Meet the Press.” He also was booked on the Spanish-language networks Telemundo and Univision. Sessions and Schumer were on ABC while McCain appeared on CNN.