After winning the position of Senate minority leader, Senator Chuck Schumer of New York slipped into his hideaway office at the Capitol on Wednesday to share the news with his parents and daughter. Credit Stephen Crowley/The New York Times

WASHINGTON — Congressional Democrats, divided and struggling for a path from the electoral wilderness, are constructing an agenda to align with many proposals of President-elect Donald J. Trump that put him at odds with his own party.

On infrastructure spending, child tax credits, paid maternity leave and dismantling trade agreements, Democrats are looking for ways they can work with Mr. Trump and force Republican leaders to choose between their new president and their small-government, free-market principles. Senator Chuck Schumer of New York, elected Wednesday as the new Democratic minority leader, has spoken with Mr. Trump several times, and Democrats in coming weeks plan to announce populist economic and ethics initiatives they think Mr. Trump might like.

Democrats, who lost the White House and made only nominal gains in the House and Senate, face a profound decision after last week’s stunning defeat: Make common cause where they can with Mr. Trump to try to win back the white, working-class voters he took from them, or resist at every turn, trying to rally their disparate coalition in hopes that discontent with an ineffectual new president will benefit them in 2018.

Mr. Trump campaigned on some issues that Democrats have long championed and Republicans resisted: spending more on roads, bridges and rail, punishing American companies that move jobs overseas, ending a lucrative tax break for hedge fund and private equity titans, and making paid maternity leave mandatory.

Some Democrats are even co-opting Mr. Trump’s language from the campaign. “Every single person in our caucus agrees the system is rigged,” said Senator Debbie Stabenow, Democrat of Michigan.

Still, there will be areas of bright-line disagreement. Democrats are speaking out against Mr. Trump’s appointment of Stephen K. Bannon as his chief strategist, and will oppose his promised tax cuts for the wealthy and his vow to deport millions of illegal immigrants.

What is not clear is whether Mr. Trump will hew to his stated agenda or turn it over to Republican lawmakers who seek a far more traditional conservative program.

In several key states, namely Pennsylvania, Ohio and Wisconsin, Democratic Senate candidates and the party’s presidential nominee, Hillary Clinton, failed to connect with middle-class voters, who gravitated instead to the populist appeals of Mr. Trump.

“There is an acknowledgment that it is very shortsighted to blame this loss on a letter from the F.B.I. or what states Hillary went to,” said Senator Amy Klobuchar, Democrat of Minnesota. “We need to do a better job having a bold sharp focus on the economy,” she said, like the cost of prescription drugs and predatory pricing on foreign steel.

“It is not just being a collection of groups,” she added. “It’s talking about policies in a serious way and talking about them in a way that touches all Americans.”

The competing political forces were evident in Mr. Schumer’s selection of a leadership slate that reflects competing strains within the party, from Senators Bernie Sanders of Vermont and Elizabeth Warren of Massachusetts, the best-known figures in the progressive wing, to Senator Joe Manchin III of West Virginia, one of a half-dozen moderate Democrats up for re-election in 2018.

“This team is ideologically and geographically diverse, it mixes the wisdom of experience with the vigor of youth, at least in Senate years,” Mr. Schumer said.

But Mr. Schumer’s immediate challenge will be to meet the often competing imperatives of those senators, who reflect the Democrats’ larger struggle of whether to try to tailor an appeal to the working-class white voters who defected to Mr. Trump or to try to increase the so-called Obama coalition anchored by minority and younger voters.

That debate is playing out in some ways in the House, where Democrats have been knotted up in an internal battle over whether Representative Nancy Pelosi of California, who hails from one of the wealthiest, most liberal districts in the nation, ought to make room at the top for a new leader, possibly from a Rust Belt state.

Representative Tim Ryan of Ohio, a former football player from the ailing industrial region around Youngstown, is talking about challenging Ms. Pelosi, and the contrast he would present would be stark.

Democrats need someone “like me, who has constituents and friends who are steelworkers or work in construction,” Mr. Ryan told The Youngstown Vindicator. “The economy and blue-collar jobs are important for us as a party. We need leaders who can go into these Great Lakes districts.”

The struggle to stitch together a winning coalition will play out in the competing policy ideas that Democrats are offering.

In a speech this week to the A.F.L.-C.I.O. Executive Council, Ms. Warren, who sparred in viciously personal terms with Mr. Trump during the campaign, noted the many ways she agreed with the president-elect.

“He spoke of the need to reform our trade deals so they aren’t raw deals for the American people,” she said. “He said he will not cut Social Security benefits. He talked about the need to address the rising cost of college and about helping working parents struggling with the high cost of child care. He spoke of the urgency of rebuilding our crumbling infrastructure and putting people back to work. He spoke to the very real sense of millions of Americans that their government and their economy has abandoned them. And he promised to rebuild our economy for working people.”

Senator Sherrod Brown, Democrat of Ohio, long a critic of trade deals, said in an interview that he had spoken extensively with Mr. Trump’s trade adviser and would work with him on issues concerning steel workers. “We can work with him on things we agree on,” Mr. Brown said. “On Bannon, no.”

Mr. Brown sent a letter to Mr. Trump urging him to renegotiate the North American Free Trade Agreement, withdraw from the Trans-Pacific Partnership, make changes to the trade relationship with China and fight currency manipulation, which is also a pet issue of Mr. Schumer. Mr. Sanders put out a statement after the election saying he too would work with Mr. Trump on areas of populist agreement.

At the same time, they remain his adversary on other matters. For example, Senator Dianne Feinstein of California, now the highest-ranking Democrat on the Senate Judiciary Committee, said in a statement on Wednesday that “the committee will pay very close attention to proposed nominees to ensure the fundamental constitutional rights of Americans are protected.”

Republicans will not like many of these proposals, but they have been fulsome in their praise of Mr. Trump since his election. Speaker Paul D. Ryan, for instance, has repeatedly said that he expects Mr. Trump to work with Republicans on their agenda of rolling back the Affordable Care Act and making large-scale changes to the tax code and entitlements.

While Mr. Trump’s policies are one matter, his nominations to his cabinet and other senior positions are another, and there, Senate Republicans who will vote on their confirmations are in a quandary. Just as they were asked to comment on Mr. Trump’s more incendiary statements during the campaign, they are now being asked to weigh in on contentious names being floated for high positions. Republican senators like John McCain of Arizona also have been implicitly critical of Mr. Trump’s cozy views toward Russia.

On one area, though, Democrats and Republicans agree: Most of them strongly oppose Mr. Trump’s proposal for congressional term limits.

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