House Speaker Paul D. Ryan (R-Wis.) wasn’t invited to the president’s dinner with Democratic leaders this week. (J. Scott Applewhite/AP)

President Trump's latest tendency to turn to Democrats to hash out major legislative deals has left Republican leaders facing a new reality as a daunting fall agenda looms: They are at their lowest moment of influence this year.

Despite their control of both chambers and with a GOP partner in the White House, congressional Republicans are laboring, sometimes awkwardly, to project leverage over efforts to rewrite the nation's tax laws and craft a bill to decide the fate of hundreds of thousands of young undocumented immigrants.

Some are privately fuming over the valuable political cover Trump is giving to centrist Democratic senators who are top targets in the 2018 midterms in states the president won. By negotiating with them and appearing at events together, the president is potentially easing their challenge of winning conservative voters.

Republicans have played down Trump's talks with Democrats, issued warnings that the effort could prove futile and looked for a silver lining — that the president is taking the politically risky lead in shepherding legislation on divisive matters.

But so far, none of these approaches have produced what GOP leaders on Capitol Hill hoped they would have after their party won the White House and Congress in November: control.

"It's not so much their power as their ability to influence the president," said former Republican National Committee chairman Michael Steele. Trump, he said, is "recognizing: 'I don't just have to play with you, Paul and Mitch. I get to play with Chuck and Nancy as well.' "

Senate Majority Leader Mitch McConnell (R-Ky.) has been overruled by President Trump in some talks with Democrats. (J. Scott Applewhite/AP)

"Chuck and Nancy" are Senate Minority Leader Charles E. Schumer (D-N.Y.) and House Minority Leader Nancy Pelosi (D-Calif.), who over dinner with Trump on Wednesday night agreed to work on a deal to save from deportation young undocumented immigrants who were brought to this country as children.

"Paul and Mitch," more formally House Speaker Paul D. Ryan (R-Wis.) and Senate Majority Leader Mitch McConnell (R-Ky.), were not invited. Trump called each of them Thursday morning to catch them up — well after the rest of the country learned the news.

There has been considerable dispute since the dinner about what, exactly, Trump and the Democrats agreed to over their meal of honey sesame crispy beef and chocolate cream pie.

To hear Republican leaders on Capitol Hill tell it, whatever it was, it was no big deal.

It was merely a "deal to make a deal," explained Sen. John Cornyn (R-Tex.), McConnell's top deputy. Or "an agreement to agree," he said a few minutes later, finding a different way to say the same thing.

On the other side of the Capitol, Ryan had to deal with questions at a news conference where he would have preferred to deal with queries about the funding bills he was touting.

"Have you asked the president to at least check with you before he makes an agreement with Democrats?" one reporter asked.

Ryan chuckled through the question after sipping from a water cup.

"First off, there's no agreement," he replied. "The president and the chief of staff called me from Air Force One today to discuss what was discussed. And it was a discussion, not an agreement or a negotiation."

Regardless of the label, Trump had sent a clear message: For the moment, at least, McConnell and Ryan have been stripped of much of the deference presidents historically invest in their party's leaders on Capitol Hill.

The president's sudden desire to negotiate with Schumer and Pelosi — they also struck a deal last week to raise the debt ceiling and keep the government from shutting down, overruling the terms McConnell and Ryan had pushed — comes after an unproductive eight months when Republicans mostly relied on their own ranks.

It didn't work, most pointedly in the failed effort to repeal and replace the Affordable Care Act. After that, some GOP lawmakers said, it was no surprise that Trump was trying a new approach. Some even suggested that the strategy is refreshing.

"To me, the power to lead is the power to persuade," said Sen. Ron Johnson (R-Wis.), who took part in a bipartisan dinner with the president this week. "And we need to do a little more persuasion. As Republicans, we need to win more political arguments. Rather than try to muscle a vote, we ought to come up with proposals and find out what works."

Persuading Democrats to support Republican ideas on tax reform, GOP leaders say, will be very challenging. In the Senate, McConnell's allies have consistently raised this point — as if to signal to the president that his time may be better spent locking down support in his own party.

Sen. John Thune (S.D.), the chamber's third-ranking Republican, said he was "skeptical" that many Democrats would "want to vote for a tax bill that the White House or the Republicans up here write."

"We'll see," he concluded.

But there are questions about how invested Trump is in traditional Republican ideas. He has shown a willingness to stray from GOP orthodoxy before. In the tax reform debate, he's doing it again.

In his public appearances and on Twitter, Trump has promised a historic tax cut without acknowledging the difficult trade-offs necessary to close loopholes and offset the revenue lost through major rate reductions. A failure to pay for those lower rates may mean that the cuts could be only temporary — dashing one of Ryan's central goals.

Trump also reportedly told Democrats that he would not cut taxes on the rich.

Earlier this week, Ryan appeared to back away from his long insistence that the tax plan would not cut government revenue, thus adding to the federal budget deficit but potentially averting the need to make tough choices. A Ryan spokeswoman said afterward that his remarks did not reflect a change of position.

Ryan allies believe that Trump, by cozying up to Democrats, may have done the speaker a favor to help him manage the fractious conservative wing of his conference — giving the speaker a rare chance to defend hard-right priorities against Trump's machinations.

Senior Republican Senate aides, meanwhile, insisted this week that McConnell and his top deputies control what comes to the Senate floor and when — notwithstanding what the president demands.

Responding to Trump's immigration meeting, McConnell issued a short statement in which he put a not-so-subtle onus on the president to make the next move: "We look forward to receiving the Trump administration's legislative proposal as we continue our work on these issues."

On the other hand, some McConnell allies see the president setting himself up to be primarily responsible for — and taking credit for — progress on immigration, an issue that has divided lawmakers for decades. And they welcome that.

"That's a fundamentally different place for Congress to be from where they have been," said Josh Holmes, a former McConnell chief of staff.

Beyond the difficulty of the legislative debates, Trump's outreach to centrist Democrats such as Sens. Joe Manchin III (W.Va.) and Heidi Heitkamp (N.D.) has put Senate Republicans on more perilous terrain heading into 2018.

A few Republicans said they view some of Trump's gestures as over the top. At one recent tax event in North Dakota that stuck out for many of them, Trump invited Heitkamp onstage and called her a "good woman." Heitkamp is up for reelection next year, and McConnell and Republicans view her seat as a prime pickup opportunity.

Such moves have generated frustration at the National Republican Senatorial Committee, according to two Republicans familiar with the situation who spoke on the condition of anonymity to describe private sentiments.

McConnell entered this election cycle looking at a map ripe for gains to pad his 52-to-48 majority, with many more Democrats than Republicans defending competitive seats — and many in states Trump won. But now, fundraisers, strategists and other Republicans with a close eye on Senate races are complaining about candidate recruiting woes they think Trump is fueling, intentionally or not.

"People are by and large incredibly pissed," said one of the Republicans familiar with the situation.

Congressional Republicans are coming to terms with the possibility that they may have to deal with Trump leaning into his Democratic outreach for an extended period of time. The president has shown no signs that he will back off his new strategy anytime soon.

"Many Republicans really like it," he told reporters on Air Force One on Thursday.

He added: "I'm a Republican through and through, but I'm also finding that sometimes to get things through, it's not working that way."

Ed O'Keefe contributed to this report.