As the new guy in town with an agenda, President Trump is waging a charm offensive unseen in Washington in years, an effort that is galling Democrats and beguiling some Republicans who opposed his candidacy.

In the past two weeks, the president and first lady Melania Trump have hosted intimate White House dinners for onetime presidential rivals Sens. Marco Rubio of Florida and Ted Cruz of Texas, who were mocked by Mr. Trump during the GOP primary as “Little Marco” and “Lyin’ Ted.”

The president also held a private lunch with Sen. Lindsey Graham, South Carolina Republican, in the presidential dining room adjacent to the Oval Office. Mr. Graham said afterward that Mr. Trump “is in deal-making mode, and I hope Congress is like-minded.”

Lawmakers are visiting the White House for bowling parties — Budget Director Mick Mulvaney has invited members of the conservative House Freedom Caucus for pizza and bowling at the White House lanes this week. Many of the two-dozen lawmakers oppose the administration-backed measure to repeal and replace Obamacare, saying it doesn’t go far enough to undo government mandates and Medicaid expansion.

Mr. Trump’s practiced art of persuasion has become an affront to some Democrats, such as House Minority Leader Nancy Pelosi of California.

“I find the charm offensive offensive. I really do,” Mrs. Pelosi said Friday during a breakfast hosted by The Christian Science Monitor. “If he’s in a deal-making [mood], does that mean bowling at the White House in return for support for legislation? I think he’s making fools of his own people.”

Such personal outreach to Congress was a glaring weakness of former President Barack Obama, who disliked schmoozing and found even fewer reasons to try after his so-called fiscal “grand bargain” with Republican leaders fell apart in 2011. He also said he didn’t want to neglect the responsibilities of fatherhood.

“Sometimes Michelle and I not doing the circuit and going out to dinners with folks is perceived as us being cool,” Mr. Obama said in 2012. “It actually really has more to do with us being parents.”

For Mr. Trump, who has said it wasn’t unusual for him to make more than 100 phone calls per day as CEO of the Trump Organization, banter and persuasion are second nature.

“The president is a man of boundless energy, optimism,” Vice President Mike Pence said Saturday.

Not that Mr. Trump’s outreach is always succeeding, however. On Thursday, just two days after he met in the Oval Office with Mr. Trump, Sen. Tom Cotton, Arkansas Republican, criticized the administration-backed health care bill on Twitter, saying it “can’t pass Senate w/o major changes.”

The senator’s early-morning tweets caused irritation in the West Wing, where White House counselor Kellyanne Conway handed the president a handwritten note in bold black letters during an Oval Office staff meeting to call Mr. Cotton about the development.

White House press secretary Sean Spicer said Mr. Trump isn’t beholden to political ideologies in his efforts to get things done, whether it’s replacing Obamacare or smoothing over tensions from last year’s campaign.

“This is a president that’s going to engage with everybody that can help join in proposing ideas and thoughts and opinions on how to move the country forward,” Mr. Spicer said.

Perhaps one of the best examples of that attitude so far came Wednesday night, when Mr. Trump hosted Alaska’s Republican senators, Lisa Murkowski and Dan Sullivan, for an Oval Office meeting with Interior Secretary Ryan Zinke.

Both senators had abandoned Mr. Trump in the presidential race in October after a decade-old audio tape surfaced of Mr. Trump making crude comments about women. Ms. Murkowski said she was “disgusted” and called on Mr. Trump to drop out of the race; Mr. Sullivan said the comments were “reprehensible” and withdrew his endorsement of the Republican nominee.

Their meeting with Mr. Trump was initiated by the White House. Ms. Murkowski came armed with maps of Alaska to show the president the state’s challenges with transportation, energy exploration and a stubborn recession.

The president surprised the lawmakers by talking about his grandfather Frederick Trump’s time spent in the Yukon in the early 1900s.

“When you have that kind of very real connection — I think it makes people excited,” Mr. Sullivan told the Alaska Dispatch News later. “He seemed very enthusiastic about the connection with his grandfather — you could see that was very sincere.”

Ms. Murkowski, who also voted against the nomination of Education Secretary Betsy DeVos, told ADN that Mr. Trump “was very generous in his comments, and very kind.”

“We had a very good and very cordial and, most importantly, a very productive discussion,” she said. “I think he’s looking at this as a good, positive relationship.”

A key topic of discussion in the meeting was oil-and-gas drilling and options for addressing the Obama administration’s eleventh-hour moves to prevent drilling in the Arctic. Ms. Murkowski expressed satisfaction that the administration intends to be “a willing partner” on energy exploration.

She said they didn’t discuss their differences from the campaign, although she had told the president in an earlier phone call that she intended to work with him.

The legislation to repeal and replace Obamacare now stands as the first big test of Mr. Trump’s powers of persuasion. While the president is showing he can woo lawmakers, he also will play hardball — planning trips to various states to sell the plan directly to voters. On Wednesday the president will hold a campaign-style rally in Nashville, Tennessee, where he’s expected to promote the plan.

Republican strategist Kristen Soltis Anderson said the president’s sales job on health care could present a quandary for conservative lawmakers who are opposing the plan.

“There are a lot of these folks in the House Freedom Caucus who would love nothing more than to go back to their districts to say they stuck to it the man, they stuck it to the establishment, they stuck it to leadership,” Ms. Anderson said Sunday on ABC’s “This Week.”

“But if it means sticking it to President Trump, I think that changes the calculation. These folks in House Freedom Caucus districts wound up having Trump win by an average margin of 25 points. These are places were Donald Trump is very popular. And so if he gets behind this fully, if this is ‘Trumpcare,’ it’s going to be a lot harder for them to vote against it.”

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