President-elect Donald Trump walks through the U.S. Capitol with his wife, Melania, and Senate Majority Leader Mitch McConnell (R-Ky.) on Nov. 10. (Bill O'Leary/The Washington Post)

It’s one of the most exclusive clubs in Washington, packed with mostly older, wealthier members and governed by a web of byzantine rules and backroom dealmaking. It’s the kind of place that business executive Donald Trump might enjoy — and that President-elect Donald Trump says he abhors.

It’s the U.S. Senate, a place critical to enacting Trump’s sweeping agenda of repealing the Affordable Care Act, overhauling the nation’s tax code, renegotiating global treaties and spending potentially trillions of dollars to build roads and bridges. But few senators will agree with everything Trump wants and fewer actually understand the incoming president. The only one who does, Sen. Jeff Sessions (R-Ala.), is poised to serve as attorney general.

“Look, Donald Trump’s never been in public office before. This is going to be a whole new experience for him, and we’re all going to have to learn to adjust and grow from it,” said Sen. John Cornyn (Tex.), the chamber’s second-ranking Republican.

Trump’s on-again, off-again relationship with House Speaker Paul D. Ryan (R-Wis.) has been under constant scrutiny, and even after Trump’s surprise victory, many insiders focused on how the 70-year-old billionaire could work with the 46-year-old speaker. But if the recent past is any indication, the dynamic to watch could actually be between Trump, the self-described master negotiator, Senate Majority Leader Mitch McConnell (R-Ky.) and Sen. Charles E. Schumer (D-N.Y.), the incoming minority leader.

An agriculture metaphor advanced by Sen. Roy Blunt (R-Mo.) — who survived a tough reelection battle by sticking close to Trump — may perfectly capture this brave new Washington.

Sen. Lindsey O. Graham (R-S.C.) speaks with reporters in the U.S. Capitol on Nov. 15, 2015. On Trump he warns colleagues, “Don’t stereotype him and be open-minded.” (Cliff Owen/AP)

Blunt compared the fledgling power structure to a “hybrid vigor,” when animals are crossbred to create a superior species.

“You might have a hybrid vigor in the government that Trump brings that wouldn’t be there otherwise. And hybrid vigor is usually a good thing,” he said, adding later that the McConnell-Schumer-Trump combination “might be a really significant thing.”

The old dynamic saw McConnell often pulling out last-minute solutions — with the help of an elder Democratic statesman — to myriad governing crises when the House lobbed an unpassable bill to the upper chamber.

On the surface, things may not be that different. McConnell still controls a Republican Senate, and Democrats have the power to block objectionable pieces of the Trump agenda and Supreme Court nominees. Republicans could push through key pieces of Trump’s platform by using fast-track rules for fiscal policy items — although enacting other pieces would still require 60 votes in the Senate.

It’s a situation similar to the first years of the Obama presidency, when a Democratic Congress and president muscled through many of their priorities despite a vow by McConnell to stop them and make Obama a one-term president.

But Trump’s agenda — which includes a big-spending infrastructure program and preserving Medicare and Social Security — may find more support among Democrats than Republicans.

Senate Majority Leader Mitch McConnell (R-Ky.) on Nov. 10 said he had a “first-class meeting” with President-elect Donald Trump. (The Washington Post)

And Democrats say, at least for now, that they will approach things differently.

“We will work with President Trump when he agrees with our positions and values, but when he goes against our values and our positions, we’ll fight him with everything we have,” Schumer said Friday in an interview.

The New York senator said that he has spoken with Trump “two or three” times since the election in calls initiated by the president-elect. Trump aides didn’t return requests for comment, and Schumer declined to specify what was discussed.

He said Democrats are eager to work with a version of the president-elect that has yet to manifest itself — a candidate who vowed to “drain the swamp” and upend the old ways of Washington and who as president-elect is interested in preserving key elements of Obamacare.

Schumer said that he hopes Trump “will not just hew to the hard-right line and realize that he won the election by attacking both the Democratic and Republican establishments. My greatest worry is that he’ll just adopt a hard-right line. I think it would be bad for America and, frankly, make his presidency a flop.”

Trump tweeted on Sunday morning: “I have always had a good relationship with Chuck Schumer. He is far smarter than Harry R and has the ability to get things done. Good news!”

McConnell, who aides said is in touch frequently with Trump or his top aides, vowed in recent days to “address the real concerns of the American people, not go back and re-litigate what anybody on either side may have said during a very hotly contested presidential race.”

McConnell said that he hopes to quickly take steps to repeal President Obama’s health law, revamp the nation’s tax system, install a conservative Supreme Court justice and fill other federal judicial vacancies. Energy is another priority of McConnell’s — he has urged Trump to rapidly approve the Keystone XL pipeline and to reverse Obama-era policies that Republicans consider a “war on coal.”

Everything on the Republican wish list is likely to face some measure of Democratic opposition. But pressuring Democrats will be easier in the next two years given the 2018 political map, when 25 of the 48 members of the Senate Democratic caucus face reelection — including 10 senators from states that Trump won. Just eight Republican seats will be up for grabs.

That dynamic means McConnell should be able to pressure several vulnerable Democrats to side with Republicans, including Sens. Joe Donnelly (Ind.), Heidi Heitkamp (N.D.), Joe Manchin III (W.Va.), Claire McCaskill (Mo.) and Jon Tester (Mont.).

Manchin sounds especially eager to cooperate with the new Republican president. Some Senate aides speculated that he earned a spot on Schumer’s leadership team partly to assuage rumors that he might switch parties — a charge that Manchin and senior Senate aides denied.

“If President-elect Trump comes with good policies, I’m going to be 1,000 percent behind him. Okay?” Manchin said. “Maybe the rest of my caucus will not, but I’m going to find a pathway forward.”

McCaskill, a vocal supporter of Obama and Hillary Clinton, remains more skeptical of engaging with Trump.

“The big question that none of us can answer right now is what role will Donald Trump’s personality play in how he governs?” she asked. “Can he sublimate his natural tendencies to make it all about him, to work across the aisle on things that we agree with on, or to unite the Republican Party?”

McConnell and Schumer epitomize the old-school way of doing Washington business: Their first instinct is to enhance their side’s chances at victory in the upcoming election. When pushed to the brink, they tend to craft complex deals averting massive crises, and their political calculations are aimed at flexibility in selling the deal to partisans.

The classic example is the $700 billion bailout of big banks in 2008. Both senators played key roles in drafting the package, which at first failed in the House. Although the overall package was unpopular, it came with a few caveats designed to protect incumbents who voted for it — all $700 billion, for instance, was not handed over at once, but instead came in tranches that Congress could impede.

But that old-fashioned dealmaking may no longer fit Trump’s Washington, and McConnell may be in the toughest spot of all.

On Friday, Stephen K. Bannon, set to serve as Trump’s top White House strategist, told the Hollywood Reporter that he is pushing the new president to adopt “a trillion-dollar infrastructure plan.” Earlier in the week, McCaskill warned that Trump’s support for infrastructure spending would ultimately cause more heartburn for Republicans given that they fought such costly measures during Obama’s presidency.

“All we’ve heard from the Republicans is we can’t do a big spending bill unless it’s paid for, and I don’t think Donald Trump is much interested in paying for anything,” McCaskill said. “He loves OPM — ‘Other People’s Money,’ including taxpayers’.”

Schumer added that congressional Republicans would “rue the day” they try to repeal Obamacare.

“Why? Because all of a sudden they’re realizing there’s a lot of good things” in the law, he said, citing Trump’s openness to keeping protection for young people and those with preexisting conditions.

“There’s no way to keep the good things without keeping ACA,” Schumer said. Republicans “are a dog who caught the bus.”

But Sen. Lindsey O. Graham (R-S.C.), who as a failed presidential candidate got to know Trump, warned senators to tread carefully.

“Don’t stereotype him and be open-minded,” he said. “You’re missing who he is if you go in with a bunch of preconceived notions. He’s one of the most charming people I’ve met. And he actually can be quiet and absorb stuff.”

McConnell engaged very little with Trump during the campaign and for weeks avoided any public comments on him.

Trump and Schumer are among the most media-savvy New Yorkers, but they don’t have a close relationship. During an October interview, Schumer recalled that as a Brooklyn-area congressman in the 1980s and 1990s, he had little interaction with Trump. If the business executive ever called his congressional office, Schumer said, it was probably about Trump Village, a six-building complex for middle-income New Yorkers that was built by the president-elect’s father, Fred Trump.

During the 2008 campaign cycle, Trump hosted a fundraiser for the Democrats’ Senate campaign arm, which Schumer chaired, and helped the party raise $230,000 at Mar-a-Lago in Palm Beach, Fla.

“It was a usual fundraiser,” Schumer recalled, describing the event as “a good night” for the party’s coffers.

“People wanted to go to Mar-a-Lago, which was then a bigger deal than it is now,” Schumer added.

On Friday, Schumer said what he remembers most about Trump was “landing at LaGuardia and seeing his big plane sitting there, and I realized, what a genius — he paid the rent because it was like getting a great sign advertised because his plane said, ‘Trump.’ ”

Mike DeBonis and Anu Narayanswamy contributed to this report.