But both the White House and Senate proposals for tackling the complex and emotionally charged issue still lack key details. And potential roadblocks are already emerging over how to structure the avenue to citizenship and whether a bill would cover same-sex couples - and that's all before a Senate measure can be debated, approved and sent to the Republican-controlled House where opposition is likely to be stronger.
Obama, in the heart of the heavily Hispanic Southwest, praised the Senate push, saying Congress is showing "a genuine desire to get this done soon." But mindful of previous immigrations efforts that have failed, Obama warned that the debate would become more difficult as it gets closer to a conclusion.
"The question now is simple," Obama said during a campaign-style event in Las Vegas, one week after being sworn in for a second term in the White House. "Do we have the resolve as a people, as a country, as a government to finally put this issue behind us? I believe that we do."
Despite possible obstacles to come, the broad agreement between the White House and bipartisan lawmakers in the Senate represents a drastic shift in Washington's willingness to tackle immigration, an issue that has languished for years. Much of that shift is politically motivated, due to the growing influence of Hispanics in presidential and other elections and their overwhelming support for Obama in November.
Still, some Republicans, including House Speaker John Boehner of Ohio and Senate Minority Leader of Kentucky, responded cautiously to the proposals from the president, on Tuesday, and the Senate group, which put forward its proposals one day earlier.
"Any solution should be a bipartisan one, and we hope the president is careful not to drag the debate to the left and ultimately disrupt the difficult work that is ahead in the House and Senate," said Brendan Buck, a Boehner spokesman.
The separate White House and Senate proposals focus on the same principles: providing a way for most of the estimated 11 million people already in the U.S. illegally to become citizens, strengthening border security, cracking down on employers who hire illegal immigrants and streamlining the legal immigration system.
A consensus around the question of citizenship could help lawmakers clear one major hurdle that has blocked previous immigration efforts. Many Republicans have opposed allowing illegal immigrants to become citizens, saying that would be an unfair reward for people who have broken the law.
Details on how to achieve a pathway to citizenship still could prove to be a major sticking point between the White House and the Senate group, which is comprised of eight lawmakers - four Democrats and four Republicans.
Obama and the Senate lawmakers all want to require people here illegally to register with the government, pass criminal and national security background checks, pay fees and penalties as well as back taxes, and wait until existing immigration backlogs are cleared before getting in line for green cards. After reaching that status, U.S. law says people can become citizens after five years.
The Senate proposal says that entire process couldn't start until the borders were fully secure and tracking of people in the U.S. on visas had improved. Those vague requirements would almost certainly make the timeline for achieving citizenship longer than what the White House is proposing.
The president urged lawmakers to avoid making the citizenship pathway so difficult that it would appear out of reach for many illegal immigrants.
"We all agree that these men and women have to earn their way to citizenship," he said. "But for comprehensive immigration reform to work, it must make clear from the outset that there is a pathway to citizenship."
"It won't be a quick process, but it will be a fair process," Obama added.
Another key difference between the White House and Senate proposals is the administration's plan to allow same-sex partners to seek visas under the same rules that govern other family immigration. The Senate principles do not recognize same-sex partners, though Democratic lawmakers have told gay rights groups that they could seek to include that in a final bill.
John McCain of Arizona, who is among the eight in the Senate immigration group, called the issue a "red flag" in an interview Tuesday on "CBS This Morning."
Washington last took up immigration changes in a serious way in 2007, when then-President George W. Bush pressed for an overhaul. The initial efforts had bipartisan support but eventually collapsed in the Senate because of a lack of GOP support.
Cognizant of that failed effort, the White House has readied its own immigration legislation. But officials said Obama will send it to the Hill only if the Senate process stalls.
Most of the recommendations Obama made Tuesday were not new. They were included in the immigration blueprint he released in 2011, but he exerted little political capital to get it passed by Congress, to the disappointment of many Hispanics.
Some of the recommendations in the Senate plan are also pulled from past immigration efforts. The senators involved in formulating the latest proposals, in addition to McCain, are Democrats Charles Schumer of New York, Dick Durbin of Illinois, Robert Menendez of New Jersey and Michael Bennet of Colorado, and Republicans Lindsey Graham of South Carolina, Marco Rubio of Florida and Jeff Flake of Arizona.
Also Tuesday, in another sign of Congress' increased attention to immigration issues, a group of four senators introduced legislation aimed at allowing more high-tech workers into the country, a longtime priority of technology businesses. The bill by Republicans Rubio and Orrin Hatch and Democrats Amy Klobuchar and Chris Coons would increase the number of visas available for high-tech workers, make it easier for them to change jobs once here and for their spouses to work, and aim to make it easier for foreigners at U.S. universities to remain here upon graduation.
(Copyright 2013 by The Associated Press. All Rights Reserved.)