House Democrats are hoping to vote Friday on the next round of coronavirus relief, Majority Leader Steny Hoyer Steny Hamilton HoyerHouse Democrats postpone vote on marijuana decriminalization bill Democrats scramble on COVID-19 relief amid division, Trump surprise The Hill's Morning Report - Sponsored by The Air Line Pilots Association - Pence lauds Harris as 'experienced debater'; Trump, Biden diverge over debate prep MORE (D-Md.) said Wednesday, but disagreements over the process could push it to next week.

"Hopefully we are getting closer to an agreement, but I can't guarantee that we're going to get an agreement that we can pass on Friday. That would be optimal if we could," Hoyer told reporters on a conference call.

His comments come as party leaders negotiate a fourth coronavirus relief package, which failed last week on the Senate floor over a dispute about the size and scope of the legislation.

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Republican leaders, at the request of Treasury Secretary Steven Mnuchin Steven Terner MnuchinLawmakers fear voter backlash over failure to reach COVID-19 relief deal United Airlines, unions call for six-month extension of government aid House Democrats plan to unveil bill next week to avert shutdown MORE, are pushing for $250 billion in additional funding for the popular Paycheck Protection Program (PPP), which aims to help small businesses stay afloat through the crisis without firing employees.

Democrats, while supporting the funding boost to the $349 billion program, are insisting that it come with stipulations protecting smaller and minority-owned shops, which tend to have more tenuous relationships with banks and other lending institutions. They're concerned the bulk of the PPP funds will go to larger businesses that have well-established relationships with lenders.

"The most vulnerable are finding it the most difficult," Hoyer said.

Led by Speaker Nancy Pelosi Nancy PelosiSunday shows preview: Justice Ginsburg dies, sparking partisan battle over vacancy before election Trump is betting big on the suburbs, but his strategy is failing 'bigly' Trump orders flags at half-staff to honor 'trailblazer' Ginsburg MORE (D-Calif.) and Senate Minority Leader Charles Schumer Chuck SchumerVideo of Lindsey Graham arguing against nominating a Supreme Court justice in an election year goes viral Graham signals support for confirming a Supreme Court nominee this year Pelosi orders Capitol flags at half-staff to honor Ginsburg MORE (D-N.Y.), Democrats are pressing for language ensuring that at least $60 billion of the $250 billion in additional small-business funds go through Community Development Financial Institutions — nonprofits that cater to low-income communities.

"I will not allow anything to perpetuate the disparity and access to capital that exists in our country," Pelosi said Tuesday night in an interview with MSNBC's Chris Hayes. "So, when they said first-come, first-serve, ‘Oh we’re just serving the customers we know at the bank’ — what happens to our underbanked folks?"

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Democrats are also pressing for an additional $150 billion for state and local governments; $100 billion for hospitals and community health centers; and a 15 percent increase in food stamps as part of the interim package.

Hoyer said he's spoken to Republicans who support the Democrats' proposed additions — "so I don't think it's the substance of our request that seemed to be the problem," he said Wednesday.

"I think it was the process; they didn't like adding onto their request," Hoyer said. "But clearly the Congress is designed by the Constitution to provide resources that we believe are necessary, and the executive to carry it out. ... We would hope that we could reach agreement."

Adding pressure on lawmakers to reach a deal, existing PPP funding could be exhausted by the end of the week.

The Senate is scheduled to hold a pro forma session Thursday, providing a window for any agreement to move through the upper chamber, while the House has its next session scheduled for Friday.

Heading into that session, Hoyer took one option off the table, rejecting the idea that the House would adopt remote voting to move the next round of coronavirus relief — a strategy being pushed by some rank-and-file members amid concerns over travel and gathering in crowds.

"We are not going to be able to do that immediately, because we need to change the rules and we need to create [bipartisan] agreement," Hoyer said.