Top aides to President Donald Trump on Sunday raised the possibility of the Congress passing piecemeal coronavirus relief legislation - an approach opposed by Democrats - with a deadline looming on the expiration of enhanced unemployment benefits.

A day before Senate Republicans were due to introduce a $1 trillion coronavirus relief bill that includes reduced federal unemployment benefits, U.S. Treasury Secretary Steve Mnuchin said he believes the party can work quickly with Democrats on getting legislation passed.

Mnuchin and White House Chief of Staff Mark Meadows floated the idea of Congress passing a bill focusing on items including federal unemployment benefits that expire on Friday, while leaving other issues to be tackled later.

Leaders in the Senate, controlled by Trump's fellow Republicans, and White House officials were scrambling to hammer out an agreement on their relief proposal, which hit snags after Majority Leader Mitch McConnell announced plans to introduce the measure last week.

The Democratic-controlled House of Representatives passed a $3 trillion coronavirus relief bill in May. House Speaker Nancy Pelosi, the top Democrat in Congress, criticized Republicans for not being able to agree on a plan when Americans are "on the brink" and going hungry.

"They're in disarray and that delay is causing suffering for America's families," Pelosi told CBS's "Face the Nation."

Pelosi has been awaiting the Republican proposal so negotiations can begin on a final bill to be passed by Congress.

"So we have been ready for two months and 10 days. I've been here all weekend hoping they had something to give us," Pelosi said.

The most pressing issues are renewing the enhanced unemployment benefits - currently $600 a week - and liability protections from coronavirus-related lawsuits such as "frivolous" litigation against schools and universities, Mnuchin said.

If lawmakers can agree on these two issues, Meadows said, they can debate the rest of the relief bill in the coming weeks.

"Honestly I see us being able to provide unemployment insurance, maybe a retention credit to keep people from being displaced or brought back into the workplace, helping with our schools," Meadows told ABC's "This Week."

"If we can do that along with liability protection, perhaps we put that forward, get that passed - as we can negotiate on the rest of the bill in the weeks to come."

The coronavirus pandemic has hammered the U.S. economy and triggered large increases in unemployment. Congress is working toward passing the latest in a series of bills in response to the pandemic.

(With input from the agencies)