In addition to the legalization measures for qualifying immigrants, the proposed bill would allow certain immigrants who were deported during the Trump administration to apply to return to reunite with family or for other humanitarian reasons. The proposal also raises the number of available legal work visas.
Biden, who took office on Wednesday, faces long odds, however, to win over enough Republicans in a closely divided Congress to pass the bill, congressional aides, experts and advocates told Reuters.
The Biden administration released a summary of the bill on Wednesday, but the text - which is expected to be hundreds of pages - has not been made public.
Menendez said Biden was committed to passing immigration reform and that he would not have backed the measure if it was purely a symbolic statement of Biden's position.
"If I thought that this was just a messaging bill that was going to go absolutely nowhere, then I wouldn't have lent my name to it," Menendez said. "Having said that, yes, this will be tough."
Control of the Senate is split 50-50, with Vice President Kamala Harris as the tie-breaking vote. Democrats would need to win over 10 Republicans to avoid a "filibuster," a procedural hurdle that can delay or block legislation from coming to a vote.
Democrats control the House of Representatives, but by a narrower margin than in the previous Congress.
The challenges in the divided Senate are already on display as Democrats and Republicans negotiate the terms of their power-sharing agreement. Mitch McConnell, the Senate's top Republican, is seeking assurances that the Senate will retain the filibuster, according to his office.
White House Press Secretary Jen Psaki said during a briefing on Thursday that the administration was hopeful Biden's bill would provide "a moment of reset" that could restart discussions among lawmakers.
Biden's immigration push comes after Trump, a Republican, spent four years cracking down on legal and illegal immigration. Many Republican lawmakers backed Trump's tough immigration policies.
Republican Senator Tom Cotton of Arkansas tweeted on Monday - before Biden was sworn in - that the new president was "wasting no time trying to enact his radical immigration agenda." He called the bill "total amnesty" with "no regard for the health or security of Americans, and zero enforcement."
Senator Marco Rubio of Florida, who was involved in bipartisan immigration reform talks in 2013 before ultimately abandoning the effort, called Biden's bill a "non-starter" in a statement this week.
Senate Majority Whip Dick Durbin told reporters at the Capitol on Thursday that he had "reached out to half a dozen Republicans on immigration and they've been open to the conversation."
Democrats could have more success with narrower immigration measures tied to must-pass spending bills or pandemic relief packages than trying to attract Republican support for a broad bill, said Frank Sharry, executive director of the pro-immigrant America's Voice.
Another option would be to attempt to eliminate the filibuster or use other procedural moves to pass a bill with only 51 votes, he said.
Alex Conant, a Republican strategist who worked as a spokesman for Rubio during the immigration talks in 2013, said the bill does not appear likely to win over Republicans, but could evolve into a more palatable proposal.
Reuters/Ipsos polling show support for offering a path to citizenship for immigrants in the country illegally has risen since 2014, with 60% of Americans in favor of it in a survey in February last year.
Some advocates argue Americans are now more aware of contributions by immigrant workers because of the pandemic.
"Who stepped up to the plate when we were all locked in our houses and who put their lives on the line" said immigration attorney David Leopold, who served as a volunteer adviser to the Biden campaign. "Undocumented workers."
(Reporting by Ted Hesson in Washington and Mica Rosenberg in New York; Additional reporting by Richard Cowan in Washington, editing by Ross Colvin and Aurora Ellis) ((firstname.lastname@example.org; (646) 223-6735;))