Democrats and Republicans are already drawing red lines over what they hope will be included in the next round of coronavirus relief legislation, signaling there will be hard-fought negotiations to come on Capitol Hill.

Congressional Democratic leaders have made clear that a top priority for any upcoming aid measure will be funding for cash-strapped state and local governments whose resources have been stretched thin by the crisis.

Republicans are signaling they may be willing to accept that, but only if it comes along with conditions for liability protection that they want to see in the legislation, that Democrats are so far pushing back against.

Senate Majority Leader Mitch McConnell said on Wednesday that he is open to considering more money to help state and local governments respond to the coronavirus outbreak, but said it would be conditioned on passing liability reforms to protect employers from a rash of lawsuits he believes will be aimed at them after the pandemic clears and businesses reopen.

"We are going to insist on this reform, which is not related to money, as a condition for going forward. We need to protect the American people and protect the brave people who have been on the front lines," he told Fox News Radio.

That condition is already setting up a clash with Democrats who have been critical of the request.

Senate Minority Leader Chuck Schumer said Tuesday that by pressing for liability protections for businesses and health care professionals in the next coronavirus bill McConnell is "putting barriers in the way of giving state and local governments desperate money they need."

During a call with reporters, Schumer raised concerns that such a proposal might put workers at a disadvantage.

"I haven't seen the details of what he said, but is he saying that if an owner tells a worker they have to work next to somebody who might have coronavirus without a mask or (personal protective equipment), that that owner wouldn't be liable? That makes no sense," Schumer said.

House Speaker Nancy Pelosi similarly emphasized the importance of protecting workers when asked about the issue of liability on a call with reporters on Tuesday.

"We are there to protect the workers. I don't think at this time of the coronavirus that there is any interest in having any less protection for our workers, in fact, even more," she said.

Pelosi has also made clear that in her view, money for state and local governments is a non-negotiable.

"There will not be a bill without state and local," the speaker said last Friday.

Congress has already passed trillions of dollars in legislative relief to respond to the pandemic, but the scale of the devastation that the crisis has inflicted -- both in terms of public health and loss of life and to the economy, which has been placed in a deep freeze -- has led to both Democrats and Republicans saying more aid will be needed.

The fact that a slate of aid measures have already passed Congress may make negotiations over any upcoming aid package even more complicated given concerns among rank-and-file members on both sides over concessions made in previous packages and a sense that it is unclear how many future packages there might be to come.

That has raised the stakes of the negotiations and increased pressure on leadership on both sides of the aisle over what will be included in any next round of funding.

The Senate will return to Washington next week, despite the fact that much of the District of Columbia remains shut down in an effort to stop the spread of the coronavirus with stay-at-home order in effect until at least May 15.

The House, in contrast, will not be returning just yet. Members of the US House of Representatives will not return to Washington next week, House Majority Leader Steny Hoyer said Tuesday morning, a reversal from the plan Democratic leaders announced fewer than 24 hours earlier.

But Hoyer indicated that the House will return when it is ready to take up the next relief package, which lawmakers are referring to as "CARES 2" after the earlier passage of the more than $2 trillion CARES Act last month.

"We have decided we will not come back next week, but we will come back very soon to pass the CARES 2 piece of legislation, and at that time we will be asking members to return to the Congress, to Washington," Hoyer told reporters on a call.

CNN's Haley Byrd, Alex Rogers and Ali Zaslav contributed to this report.