Congress returns Monday for a pivotal week in which lawmakers will attempt to reach agreements on proposals for gun-control and immigration-reform -- perhaps the two biggest issues in Washington this year.

Though the Senate has been working on both issues only since the new Congress began in January, Washington lawmakers have tried for years to draft comprehensive immigration reform, while the issue of gun-control was thrust upon them after 20 first-graders were killed in a December 2012 mass shooting.

A contentious public debate over the country’s flawed immigration system is expected as a bipartisan group of eight senators finalizes a bill to secure U.S. borders, allow tens of thousands of foreign workers into the country and grant eventual citizenship to the estimated 11 million people living here illegally.

Lawmakers came close to an immigration deal on the Senate floor in 2007, but it collapsed amid interest group bickering and public backlash.

Arizona Republican Sen. John McCain, a member of the so-called “Gang of Eight,” was already warning about the sharp edges of compromise on Sunday.

"There will be a great deal of unhappiness about this proposal because everybody didn't get what they wanted," he said on CBS' "Face the Nation.”

The group planned to have a final proposal ready in March. However, talks have been slowed in part because of wrangling over the extent to which securing U.S. borders, specifically the one with Mexico, will set the stage for the path to citizenship.

New York Democratic Sen. Chuck Schumer told CBS the group hopes to reach a final deal by week’s end. He also said the legislation would be introduced in the Senate Judiciary Committee with all 50 senators voting by May.

Immigration reform is a top, second-term priority for President Obama.

On Sunday, White House adviser Dan Pfeiffer said the negotiators’ proposal is “100 percent consistent with what the president is doing so we feel very good about it."

However, he declined to say on “Fox News Sunday” whether Obama would sign legislation making a path to citizenship contingent on first securing the border.

Pfeiffer also went on TV on Sunday to garner support for the president’s gun-control proposal.

With proposed bans on semi-automatic weapons and high-capacity gun magazines off the table for now, Obama appears to be focusing his efforts on garnering public support and getting Congress to agree to universal background checks for gun buyers.

Pfeiffer said the president has “marshaled people to his side” and polls show a large majority of the public supports background checks.

“You cannot get 90 percent of the people to agree on the weather,” he told Fox. “The question is whether Congress is going to do the right thing.”

Pfeiffer said the president agrees with the efforts so far of Senate Majority Leader Harry Reid and the other senators, following the mass shootings at the Newtown, Conn., elementary school.

“This is the best response to Newtown and gun violence in the country,” he said.

The president is going Monday to the University of Hartford, in Connecticut, to talk about gun control.

A White House official told Fox News on Sunday that Obama will speak about "the obligations we have to children lost in Newtown and other victims of gun violence" and the need to act on gun-control proposals.

Senators could start debating Democratic-written gun legislation before week's end. But leaders also might decide to give negotiators more time to seek a deal on expanding background checks for firearms buyers.

Passing expanded background checks would be viewed as a victory for gun-control advocates, after Democratic leaders made it clear that supporters were nowhere close to getting a majority of votes in favor of re-instituting an assault-weapons ban.

The National Rifle Association opposes the assault-weapons ban and the expanded background checks.

Even with a background check deal, Senate debate on gun legislation may begin at a slow crawl with some conservatives promising delays and forced procedural votes.

The Senate gun legislation would toughen federal laws against illegal firearms sales, including against straw purchasers, or those who buy firearms for criminals or others barred from owning them. The legislation also would provide $40 million a year, a modest increase from current levels of $30 million, for a federal program that helps schools take safety measures such as reinforcing classroom doors.

Many experts agree that the proposal with the widest potential reach is a broadening of background checks, now required only for transactions by the roughly 55,000 federally licensed firearms dealers. Proponents want to cover private sales, such as those between individuals at gun shows or online.

One major hang-up has been Democrats' insistence on retaining records of private sales, which they say is the best way to ensure background checks are actually conducted.

The system is aimed at preventing guns from going to criminals, people with severe mental problems, some drug abusers and others.

The NRA and other critics say the checks are ignored by criminals, and they fear that expanding the system could be a prelude to the government maintaining files on gun owners.

However, Senate aides said Sunday that West Virginia Democratic Sen. Joe Manchin and Pennsylvania Republican Sen. Pat Toomey are working on a compromise to expanding required federal background checks to gun shows and to online firearms sales.

Manchin is a moderate with an A rating from the NRA, while Toomey has solid conservative credentials. The aides spoke on condition of anonymity.

With immigration reform, a major breakthrough occurred a couple of weeks ago when the U.S. Chamber of Commerce and the AFL-CIO reached a deal to allow as many as 200,000 low-skilled workers into the country each year to fill jobs in construction, hospitality, nursing homes and other industries in which employers say they have a difficult time hiring Americans.

The eight negotiators also have pledged to move the bill through the Senate Judiciary Committee, then to the chamber floor to head off complaints that the legislation is being rammed through.

Florida Republican Sen. Marco Rubio -- a potential 2016 presidential candidate who has been frustrated by efforts since being elected to the Senate in 2010 to pass immigration reform -- has emerged as a key figure in negotiations.

On Sunday, South Carolina Republican Sen. Lindsey Graham called Rubio a “game changer” for their party and assured people Rubio will not walk away from final negotiations.

“He will be there,” Graham said on NBC’s “Meet the Press.”

He also suggested, ahead of Obama releasing his 2014 budget Wednesday, Congress cannot reach a so-called “grand bargain” until it fixes immigration.

“I think if you do immigration and the grand bargain, this year will dominate the 21st century,” Graham told NBC. “The key to the grand bargain is can we solve immigration.”

Some sticking points remain -- including the plan for a program to bring in agriculture workers, who weren't included in the deal between the chamber and organized labor.

But Republicans are as interested as the president and other Democrats in reaching a comprehensive deal.

Roughly 71 percent of Hispanics voted to reelect Obama, a wakeup call to Republicans that Democrats may continue to hold the White House unless they make a better effort to bring Latino voters into the party.

The Associated Press contributed to this story.