President Biden, Democratic lawmakers and congressional Republicans all say they want to do something — anything — to upgrade the nation’s infrastructure. But they don’t yet agree on much else, including what “infrastructure” actually means.

The lingering schisms surfaced anew as Biden hosted Senate Republicans at a closely watched Oval Office meeting on Thursday. Even as both sides stressed their commitment to a bipartisan deal, they acknowledged afterward that they’re still haggling over what it should include — and haven’t even touched the fierce debate over how to pay for it.

The latest round of infrastructure talks came as the White House inched closer to its self-imposed Memorial Day deadline, a date by which it says it expects progress on advancing Biden’s jobs and infrastructure plan.

Key committees in Congress are trying to push forward transportation bills that achieve some of the president’s initiatives by then, although some of those discussions in recent days have been snarled in partisan disputes.

Opening the meeting, Biden stressed that Democrats are negotiating in “good faith” after Republicans blasted the roughly $2 trillion infrastructure package he unveiled earlier this spring as too vast in size and scope. Asked whether he might accept a smaller price tag, the president told reporters, “I’m prepared to compromise.”

Republicans similarly praised the gathering, and the two sides plan to exchange additional policy proposals entering next week. Sen. Shelley Moore Capito (R-W.Va.), who is leading the GOP’s negotiations on infrastructure, described it as “very productive” and a “more than courteous give-and-take.”

“We did talk specifics, and the president has asked us to come back and rework an offer so that he can then react to that and re-offer to us,” Capito told reporters after the meeting concluded. “So we’re very encouraged. We feel very committed to the bipartisanship that we think that this infrastructure package should carry forward.”

Beyond the platitudes, though, none of the lawmakers have been able to map out a path that will thread the political needle — satisfying enough Republicans to give Biden the cross-party achievement he craves without significantly sacrificing Democrats’ ambitions on a transformational jobs plan.

“I think we’re still looking for a bill that would be traditional infrastructure,” said GOP Sen. Roy Blunt (Mo.), one of the lawmakers who attended the meeting.

For now, the GOP counteroffer is a fraction of what Biden has proposed, although Blunt and his colleagues in recent days have indicated that they are willing to go somewhat higher on the final price tag. The White House has also yet to directly convey to GOP negotiators how low Biden is willing to go. What Republicans say they won’t do is to stretch the definition of infrastructure to include spending the White House says is necessary to revitalize the economy, such as home health care.

To bolster their ranks ahead of the discussions with Biden, Senate GOP leaders shared internal polling at a closed-door lunch earlier this week that showed voters generally preferred a scaled-back infrastructure plan, according to an official familiar with the figures.

In the poll, about 37 percent of those surveyed said they favored the Biden administration’s $2 trillion infrastructure proposal, while 47 percent said they preferred the Republicans’ targeted plan at a fraction of the cost, said the official, who spoke on the condition of anonymity to disclose internal data collected for Senate Republicans.

Another significant sticking point is how any package will be paid for. The main source of revenue the White House wants to use to finance its proposal would require unraveling parts of the 2017 tax law, which GOP senators have said is a “red line” for them and a position that top Republican leaders underscored in a separate meeting with Biden on Wednesday.

“I think both in yesterday’s meeting with the two leaders, and in this meeting, they clearly understand that would not be something we [could support],” Blunt said.

In turn, Senate Republicans have proposed raising the gas tax and implementing user fees for other modes of transportation, including electric vehicles. The White House, for its part, has said Biden opposes a gas tax hike.

Capito and Blunt were joined by GOP Sens. John Barrasso (Wyo.), Mike Crapo (Idaho), Patrick J. Toomey (Pa.) and Roger Wicker (Miss.). Vice President Harris, Commerce Secretary Gina Raimondo and Transportation Secretary Pete Buttigieg also attended.

Capito, Senate Minority Leader Mitch McConnell (R-Ky.) and virtually all other Senate Republicans have warned that they won’t touch the 2017 tax overhaul, drawing fresh criticisms from Democrats that the GOP is unwilling to compromise.

“The Republicans are saying those megacorporations that use roads and bridges and transportation systems every single day as part of their efforts to generate revenue . . . shouldn’t have to pay a penny, and their employees, middle-income workers, should have to bear the burden,” Sen. Ron Wyden (D-Ore.), chairman of the Finance Committee, said Wednesday.

From there, potential revenue options get more obscure. Crapo, the ranking Republican on the Finance Committee, said other potential ways to pay for an infrastructure package include public-private partnerships and government bonds. He also expressed some optimism that the White House could embrace some user fees, if not a gas tax increase.

As Biden engages in high-profile meetings with Republicans, lawmakers are working on advancing at least some infrastructure provisions through existing routes, such as legislation that reauthorizes the nation’s surface transportation programs. But even that — traditionally among the most bipartisan of endeavors on Capitol Hill — has seen signs of trouble.

In the Senate, Capito and Sen. Thomas R. Carper (D-Del.), chairman of the Environment and Public Works Committee, have struggled in their negotiations for a surface transportation bill, particularly on climate issues, according to people familiar with the discussions.

Republicans have declined to embrace some climate priorities advocated by Democrats that they had accepted last year when the GOP held the majority, including incentives to reduce emissions, according to one of the officials, who spoke on the condition of anonymity to disclose ongoing private negotiations.

“We’ve got some sticking points, no doubt about it,” Capito said Wednesday. She called that transportation bill the “anchor” of any broader infrastructure effort between the White House and Congress.

In his own private meeting with the president earlier this week, Carper had emphasized to Biden that any transportation legislation needs to address climate change, including in his discussions with Capito.

Carper and Democrats had secured $10 billion in climate investments from Republicans in the last Congress, and they have insisted on that as a minimum in the ongoing talks on a transportation bill.

Carper said in an interview that he and Capito are in intense negotiations over the transportation bill and stressed that it would include “strong climate provisions.”

The two senators do have a history of achieving agreement, successfully negotiating a $35 billion water package earlier this year that was widely heralded as an encouraging sign for bipartisanship.

Lawmakers set aside the new funds to fix pipes, combat water pollution and deliver federal aid to tribal and rural areas in greatest need. The deal sailed through Carper and Capito’s environment-focused committee and later was passed by an 89-to-2 vote.

“We are proving that our two parties can work together on legislation, including on some of the issues President Biden mentioned,” Senate Majority Leader Charles E. Schumer (D-N.Y.) said during the floor debate over the water legislation.

But even the much smaller water package illustrated the trade-offs that Democrats would have to embrace in virtually any infrastructure agreement with Republicans.

The $35 billion bill was a significant departure from the roughly $111 billion that Biden had proposed for improvements to the country’s water system, and his jobs plan also included more ambitious targets, including a goal to replace 100 percent of the nation’s lead pipes and service lines, along with more aggressive proposals to tackle ills including climate change.

Those kinds of sacrifices — greatly scaling back the Democrats’ ambitions at a time when they hold the levers of power in Washington — are ones others in the party aren’t willing to make.

“We appreciate the White House’s interest in reaching across the aisle to seek Republican support for overwhelmingly popular infrastructure priorities,” roughly 60 House Democrats across the ideological spectrum wrote in a letter being sent to Democratic leaders. “While bipartisan support is welcome, the pursuit of Republican votes cannot come at the expense of limiting the scope of popular investments.”

Jeff Stein contributed to this report.