Two leaders of the Senate's immigration overhaul effort said Wednesday that their group's work on a bill  that could win bipartisan approval in both chambers of Congress has picked up considerably.

In a joint appearance at a breakfast hosted by Politico's Playbook, Sen. Charles E. Schumer (D-N.Y.) and Sen. John McCain (R-Ariz.) said the eight member group will now meet every Tuesday and Thursday--and their staffs will meet Wednesdays--to quickly draft legislation by the end of March based on the blueprint they unveiled with fanfare on Monday.

Schumer described an open legislative process that's been rare in recent years, in which senators could propose amendments to the plan in the Judiciary Committee, with the Senate then spending maybe three to four weeks debating the massive package on the floor.

"There’s big enthusiasm," he said, "I think not just in our group but in the country."

McCain said the goal it to win as many as 80 votes in the Senate from both parties for a plan that would include quick legalization for the nation's estimated 11 million illegal immigrants and an eventual path to citizenship. He said that a Senate bill with a strong bipartisan vote could potentially pass the GOP-held House with a majority of Democratic votes and "significant" GOP support.

"Republicans are beginning to appreciate that if we’re going to have a meaningful dialogue with our Hispanic citizens, with Latino voters, then we’re going to have to resolve this issue. It’s just a fact," he said, explaining the remarkably rapid shift within the GOP on immigration policy.

McCain predicted that in the next election Republicans could lose his own state of Arizona, now represented by two GOP senators, if the immigration effort founders this year.

But he noted the group should avoid significant controversies that could hamper the legislation, such as inclusion of protections for same-sex immigrant couples. That protection an element of the plan that President Obama proposed in a major immigration policy address in Las Vegas on Tuesday.

"I'm telling you now: If you load this up with social issues and things that are controversial, then it will endanger the issue," he said.

Schumer and McCain bantered with each other during their joint appearance, a rapport  perhaps born of weeks of behind-the-scenes negotiations within the group, which has rotated between Schumer and McCain's offices.

The group met most recently on Tuesday, they said, and discussed one of the most nettlesome issues they have to negotiate before moving forward with a bill. The Senate framework allows illegal immigrants to seek green cards, the first step to citizenship -- but  only after the U.S. border is secured. The group considered measures that could be used to certify that enforcement has stepped up.

Both men also endorsed a new biometric Social Security card for use by all Americans seeking employment. The goal would be to thwart illegal immigration in the future by making it harder to forge a Social Security card and easier for employers to verify the legal status of their workers.

Schumer insisted the card would be used only for employment and not as a general identification. But he acknowledged differing opinions within the Senate's working group on the issue and said they had not yet decided whether to include it in the bill.

Obama also called for a tamper-proof Social Security cards in the White House immigration proposal released Monday. Civil libertarian groups and some conservatives, however, have expressed concern that the new card would wind up being uses as a national ID card.