The U.S. House of Representatives will take up by Wednesday the Senate version of the sweeping $1.9-trillion coronavirus relief package backed by President Joe Biden, Speaker Nancy Pelosi said on Monday.

Closing in on final approval of one of the biggest U.S. anti-poverty measures since the 1960s, Democrats aim to enact the massive legislation by Sunday, when enhanced federal unemployment benefits are set to expire.

The Senate passed its version of the bill after a marathon overnight vote on Saturday. The Senate version eliminated or pared back some provisions included in the House bill, which had increased the federal minimum wage to $15 an hour and extended expanded jobless assistance through August 29.

Now that it has passed the Senate, it must be approved again by the House before it can make its way to Biden's desk and be signed into law.

Pelosi told reporters at the Capitol that the timing of a vote on the House floor "depends on when we get the paper from the Senate." 

"We'd take it up Wednesday morning at the latest," she said.

Like the Senate, Democrats hold a very narrow majority in the chamber, meaning they cannot withstand many votes against the bill.

The first version of the bill passed in the House without a single Republican vote. Two moderate Democrats joined Republicans in voting against that version. One of them, Representative Kurt Schrader of Oregon, said on Monday he would now vote for the bill with the Senate changes.

"My concerns remain on the size and scope of this bill but believe the Senate changes provide meaningful relief for Oregonians in need," Schrader wrote on Facebook.

"Funding for our local governments, small businesses, schools, families, healthcare providers and an extension on unemployment benefits will be a lifeline for many," he said of the legislation.

Republicans, who broadly supported coronavirus relief early in the pandemic, have criticized the price tag of the Biden relief package.

On Friday, as the Senate vote was still underway, House Democratic lawmaker Bonnie Watson Coleman said she was "disgusted" by some of the changes in the Senate bill and questioned if she could support it. A spokesman for her office did not immediately respond to a request for comment.

But Representative Pramila Jayapal, head of the Congressional Progressive Caucus, told reporters she thought members of the group would back the legislation, which she described as "phenomenal" and in keeping with most of its members' priorities.

White House spokeswoman Jen Psaki praised the legislation at a news conference, saying that while there were some changes on the margins, it represented the "core" of what Biden originally proposed.