"Let's be very clear: having a pathway to earned legalization is an essential element. And I think that we are largely moving in that direction as an agreement," said Menendez, D-N.J.
But the package "will have the enhancement of the border security," he said, nodding to Republicans' priority to tighten borders to prevent future illegal immigration.
He also said the package would have to crack down on employers hiring undocumented workers.
Arizona Republican McCain has returned to the issue after having led a failed push to fix the nation's broken immigration system ahead of his 2008 bid for the White House.
McCain said: "What's changed is, honestly, is that there is a new, I think, appreciation on both sides of the aisle including, maybe more importantly on the Republican side of the aisle, that we have to enact a comprehensive immigration reform bill."
Despite making little progress on immigration in his first term, President Barack Obama won more than 70 percent of the Latino vote, in part because of the conservative positions on immigration that Republican nominee Mitt Romney staked out during the GOP primary. Asians, who immigrated to the U.S. in higher numbers than Hispanics in 2010, also overwhelmingly backed Obama. Latino voters accounted for 10 percent of the electorate in November.
Obama is to press his case for immigration changes during a trip to Las Vegas Tuesday: a pathway to citizenship for illegal immigrants that includes paying fines and back taxes; increased border security; mandatory penalties for businesses that employ unauthorized immigrants; and improvements to the legal immigration system, including giving green cards to high-skilled workers and lifting caps on legal immigration for the immediate family members of U.S. citizens.
In an opinion piece published online Sunday in the Las Vegas Review-Journal, Sen. Marco Rubio, also a member of the bipartisan Senate group, laid out his proposal to address the issue. The Florida Republican, son of Cuban immigrants, wrote that "significant progress" on enforcing immigration laws must be certified before unauthorized immigrants now in the country are allowed to apply for residency and "get in the back of the line."
Rep. Paul Ryan, the 2012 Republican candidate for vice president, said he backs Rubio's proposal.
"Immigration is a good thing. We're here because of immigration. We need to make sure it works," Wisconsin's Ryan said.
If Republicans fail to act, they will pay the price in elections for generations, McCain warned.
"Well, I'll give you a little straight talk: Look at the last election... We are losing dramatically the Hispanic vote, which we think should be ours for a variety of reasons," McCain said.
McCain and Menendez spoke with ABC's "This Week," Durbin appeared on "Fox News Sunday" and Ryan was on NBC's "Meet the Press."
Associated Press writer Philip Elliott contributed to this report.
Follow Michele Salcedo on Twitter at http://twitter.com/michelesalcedo.