If completed, the effort could represent a major breakthrough in the effort by President Barack Obama and his allies to restrict guns following last December's massacre of schoolchildren in Newtown, Conn.
Sens. Joe Manchin, D-W.Va., and Pat Toomey, R-Pa., could nail down an accord early this week, said the aides, who spoke on condition of anonymity to describe the private talks. With the Senate returning Monday from a two-week recess, the chamber's debate on gun control legislation could begin as soon as Tuesday, though it might be delayed if the lawmakers need more time to complete a deal, the aides said.
Expanding background checks to gun shows and online sales is one possibility that has been discussed, and the overall package, if completed, could still change, aides said. The senators are also discussing exempting transactions between relatives and temporary transfers for hunters and sportsmen, they said.
Manchin is a moderate who touts an A rating from the National Rifle Association, which has opposed Obama's gun control drive. Toomey has solid conservative credentials and was elected to the Senate two years ago with tea party support from his Democratic-leaning state.
A united front by the two lawmakers would make it easier for gun control advocates to attract support from moderate Democrats who have been wary of supporting the effort and from Republicans who have largely opposed it so far.
With conservative Republicans threatening a filibuster, Democrats will need 60 of the chamber's 100 votes to prevail. There are 53 Democrats and two Democratic-leaning independents in the Senate.
Federal background checks are currently required only for transactions handled by the roughly 55,000 federally licensed firearms dealers; private sales such as gun-show or online purchases are exempt. The system is designed to keep guns from criminals, people with serious mental problems, and some others.
After 20 first-graders and six elementary school staffers were killed at Newtown, Obama proposed applying the requirement to virtually all firearms sales. Gun control advocates consider expanded background checks to be the most effective step lawmakers could take to curb gun violence.
Also high on Congress' agenda is immigration, where a decisive moment is approaching.
Bipartisan groups in the House and Senate are expected to present legislation as early as this week aimed at securing the U.S. border, fixing legal immigration and granting legal status to millions who are in the United States without authorization. That will open months of debate on the politically combustible issue, with votes by the Senate Judiciary Committee expected later this month.
The House returns Tuesday and initially plans to consider a bill preventing the National Labor Relations Board from issuing rules until a dispute over administration appointees is resolved.
Lawmakers will also devote time to the 2014 budget that Obama plans to release Wednesday. It calls for new tax increases, which Republicans oppose, and smaller annual increases in Social Security and other government benefit programs, over the objections of many of the president's fellow Democrats.
On Monday, Obama travels to Connecticut to again make the case for gun legislation, with a speech at the University of Hartford.
"He's been working with both sides to try to get the strongest bill we can that has enforceable background checks," White House senior adviser Dan Pfeiffer said Sunday on ABC's "This Week."
Other Obama gun control priorities include banning assault weapons and ammunition magazines with more than 10 rounds. Both bans are expected to be offered as amendments when Senate debate begins, but the assault weapons ban seems sure to be defeated and the high-capacity magazine prohibition also faces difficult odds.
For weeks, Manchin has been part of an effort to craft a background check compromise, along with Sens. Chuck Schumer, D-N.Y., and Mark Kirk, R-Ill. Schumer focused his efforts on conservative Sen. Tom Coburn, R-Okla., but those talks sputtered over Schumer's insistence on—and Coburn's opposition to—requiring that records be kept of private gun sales.
"I'm still hopeful that what I call the sweet spot—background checks—can succeed," Schumer said Sunday. "We're working hard there."
Proponents say background checks and records—which are currently retained by gun dealers, not the government—are the best way to ensure that would-be gun-buyers' histories are researched. Opponents say the system is a step toward government files on gun owners and say criminals routinely skirt the checks anyway.
Asked about the potential compromise, Manchin spokesman Jonathan Kott said, "My boss continues to talk to all of his colleagues."
Toomey spokeswoman E.R. Anderson said she could provide no information.
Sen. John McCain, R-Ariz., urged fellow Republicans to allow debate to go forward without a filibuster, even as he declined to express support for a background check bill.
"The purpose of the United States Senate is to debate and to vote and to let the people know where we stand," McCain said, appearing alongside Schumer on CBS' "Face the Nation."
With or without an agreement, the Senate gun legislation would toughen federal laws against illegal firearms sales, including against straw purchasers, 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.
In addition, the gun bill contains language by Schumer expand background checks to cover nearly all gun transactions, with narrow exceptions that include sales involving immediate relatives. Even without a bipartisan deal, Schumer is expected to expand the exemptions to more relatives, people with permits to carry concealed weapons and others.