Lawmakers also passed a one-day stopgap spending bill to prevent a government shutdown at midnight Sunday.
The Associated Press reported that if the deal is finalized, it will be the largest spending measure, which combines the COVID-19 relief with a $1.4 trillion government-wide funding plan that the New York Times referred to as a $2.3 trillion behemoth."
After days of negotiations, McConnell said he was "relieved" that a deal was in the offing.
The House and Senate were expected to vote on the legislation on Monday. Before hitting the full House for a vote, the relief bill first must go through the House Rules Committee, though it remains unclear when that will be.
Fox News' Chad Pergram tweeted early Monday that the House Rules Committee could meet as early as 8 a.m. to prepare the bill for the floor.
"Timing for House debate is unclear," Pergram wrote. "But the House will go first, followed by the Senate. Senate doesn’t meet until noon."
According to congressional leaders, the agreement would establish temporary $300 per week supplemental jobless benefits and $600 direct stimulus payments to most Americans, along with a new round of subsidies for hard-hit businesses and money for schools, health care providers, and renters facing eviction.
"At long last we have a bipartisan breakthrough the country has needed," McConnell said, according to the Wall Street Journal. "Now we need to promptly finalize text, avoid any last-minute obstacles and cooperate to move this legislation through both chambers."
Special Assistant to the President Ben Williamson in a statement, "President Trump has pushed hard for months to send Americans badly needed financial relief. We look forward to Congress sending a bill to his desk imminently for signature."
House Speaker Nancy Pelosi, D-Calif., and Senate Minority Leader Chuck Schumer, D-N.Y., said in a joint statement their goal is to "crush the virus and put money in the pockets of the American people."
"Today, we have reached an agreement with Republicans and the White House on an emergency coronavirus relief and omnibus package that delivers urgently needed funds to save the lives and livelihoods of the American people as the virus accelerates," it said.
Senators held talks on Capitol Hill late Saturday when Sen. Pat Toomey, R-Pa., hit the Senate floor with a list of grievances on the $900 billion package.
But Fox News learned late Saturday that Toomey agreed to compromise language from his initial proposal that would have prevented the Federal Reserve chairman from establishing federal lending programs similar to those set up in March to help credit markets function during massive U.S. economic shutdowns.
"These new emergency lending facilities were always intended to sunset at end of this year," McConnell said on Sunday. "Sen. Toomey and our Democratic colleagues have landed on compromise language that ensures this will, in fact, happen. It also redirects the dormant money in these accounts toward the urgent needs of working Americans and ensures that these expiring programs cannot be simply restarted or cloned without another authorization from Congress, all while preserving both the independence of the Federal Reserve and proper authority of Congress."
The apparent compromise, following discussions between Toomey and Schumer, appeared to open a path to a final agreement on the coronavirus package.
The plan was to load the $900 billion coronavirus bill onto a $1.4 trillion omnibus spending plan to fund the government through Sept. 30, 2021.
Sen. John Barrasso, R-Wyo., told "Fox News Sunday" that the deal is expected to include $600 checks to most Americans, covering adults and children, extended unemployment benefits, and additional paycheck protection loans for small businesses.
After the announcement, Schumer and Pelosi offered additional details, including $25 billion in rental assistance, $15 billion for theaters and other live venues, $82 billion for local schools, colleges, and universities, and $10 billion for child care.
But Pelosi qualified that the bill was only an initial step, telling her colleagues in a letter that "more help will be on the way" when President-elect Joe Biden enters the White House.
The bill also was an engine to carry much of Capitol Hill's unfinished business, including a 400-page water resources bill that targets $10 billion for 46 Army Corps of Engineers flood control, environmental and coastal protection projects. Another addition would extend a batch of soon-to-expire tax breaks, including one for craft brewers, wineries and distillers.
House Democrats had passed a massive $3 trillion HEROES Act that Senate Republicans rejected in favor of a more targeted approach to helping American people and businesses.
Fox News' Caitlin McFall, Chad Pergram, Edmund DeMarche, Ronn Blitzer and Jason Donner contributed to this report.