Democratic and Republican leaders on Wednesday renewed their fitful efforts to impose greater order on a freewheeling presidential race and to bring to heel a pair of political renegades, Senator Bernie Sanders andDonald J. Trump, whose upstart campaigns have roiled the political establishment.

For Democrats impatient to rally around Hillary Clinton as their presumptive nominee, Mr. Sanders has lingered as an obstacle so stubborn and frustrating that only President Obama’s intervention might dislodge him soon.

Mr. Obama is expected to meet with Mr. Sanders in Washington on Thursday and increase the pressure on the irascible Vermonter to defer to Mrs. Clinton. Mr. Obama, White House aides said, intends to nudge Mr. Sanders toward embracing her, stressing that Mr. Sanders can further his policy agenda while unifying the party to defeat Mr. Trump in November.

The White House strategy will culminate with the president’s formal endorsement of Mrs. Clinton in the coming days, followed by an appearance with her on the campaign trail soon after.

But leaders in both parties are approaching Mr. Sanders and Mr. Trump gingerly, fearful that strong-arm tactics might foster resentment.

Each man is essentially being asked to shed part of his core political persona in the interest of party unity. For Mr. Sanders to step away from the roaring crowd, or for Mr. Trump to back away from a televised brawl, would represent a wrenching departure from form.

Mindful of that, Democrats want to ease Mr. Sanders from the race without alienating his supporters.

In a taped appearance for “The Tonight Show” that will air Thursday night, Mr. Obama praised Mr. Sanders, saying he brought “enormous energy” and “new ideas” to the campaign, making Mrs. Clinton better.

My hope is over the next couple weeks we’re able to pull things together,” the president said.

Vice President Joseph R. Biden Jr. said, “We should be a little graceful and give him the opportunity to decide on his own.”

Republicans face a thornier challenge in grappling with Mr. Trump, whose inflammatory comments and slapdash campaign style have alarmed party leaders throughout the race. Since securing his spot as the presumptive Republican nominee, he has made only transient efforts to impose discipline on his campaign and turn toward courting mainstream general election voters.

Mr. Trump appeared briefly chastened on Tuesday night, delivering a prepared victory speech free of vulgarity or racial attacks. But he quickly reverted to a more improvisational style, complaining in multiple interviews that Republicans had been wrong to criticize him for denouncing a federal judge because of his Hispanic heritage.

Still, House Speaker Paul D. Ryan assured colleagues in a conference on Wednesday that he remained supportive of Mr. Trump’s candidacy. Mr. Ryan had criticized him this week for using “racist” language in criticizing Gonzalo P. Curiel, a United States District Court judge overseeing litigation involving Mr. Trump.

Mr. Ryan told lawmakers that he continued to believe Mr. Trump represented the party’s best chance to enact its agenda, said Brendan Buck, a spokesman for Mr. Ryan. But Mr. Ryan, he said in an email, “also told them he’ll continue to speak out when/if necessary to protect our identity as a party.”

Steering carefully between Mr. Sanders and Mr. Trump is Mrs. Clinton, who praised Mr. Sanders and his supporters with her victory speech on Tuesday night, while delivering an aggressive denunciation of Mr. Trump that doubled as a sales pitch for the general election.

Allies of Mrs. Clinton said on Wednesday that they were confident Mr. Sanders would come around in time, and that the broad message she had outlined for the fall campaign would suffice to win over doubters on the left.

Mayor Eric Garcetti of Los Angeles, a Democrat who campaigned for Mrs. Clinton in California, predicted that the urgency of fighting Mr. Trump would bring the party together and ease the hurt feelings from the nominating contest.

“This is definitely a crisis moment,” Mr. Garcetti said. “People come together in crises, and Donald Trump is that crisis.”

Mr. Sanders has given no indication that he will end his campaign before the July convention in Philadelphia, though he collected far too few delegates to threaten Mrs. Clinton’s nomination in a floor fight.

Facing a stinging defeat in the California primary on Tuesday, Mr. Sanders responded with defiance. He declined to acknowledge that Mrs. Clinton had won enough delegates to capture the nomination and vowed to keep campaigning in the District of Columbia, which votes next week.

There have been a few early attempts at Clinton-Sanders diplomacy: Their campaign managers, Robby Mook and Jeff Weaver, spoke on Tuesday, and the Clinton campaign has been in direct contact with elected officials backing Mr. Sanders.

Senator Jeff Merkley of Oregon, the lone senator to back Mr. Sanders, said he was in close contact with Mrs. Clinton’s advisers and declared it time to close ranks for the general election.

“I will absolutely support our Democratic nominee, and that nominee is Hillary Clinton,” Mr. Merkley said in a telephone interview, “and I’ll keep working with her and her team in support of the ideas that Bernie Sanders put forward.”

Liberal activists aligned with the Sanders campaign have begun to inch away from his tenacious bid. Several members of Congress and two prominent advocacy groups on the left, MoveOn.org and Democracy for America, said on Wednesday that they would line up behind Mrs. Clinton as the candidate who has won the most pledged delegates.

Mr. Sanders offered no immediate response to the signs that his core liberal coalition might be faltering.

Activists on the right have had even less success at applying pressure to Mr. Trump. Republicans have repeatedly pleaded and cajoled Mr. Trump to run a more professional campaign, reflecting greater attentiveness to rules of political and social decorum.

Mr. Trump has essentially disregarded those demands and charted his own course as an insurgent candidate with a skeleton crew of campaign staff.

A new test of Mr. Trump’s campaign begins on Thursday as the candidate and his top aides embark on their most concerted push yet to win over big Republican donors. On Thursday, 70 donors are to meet at Trump Tower in New York, followed by lunch at the Four Seasons hotel. Mr. Trump is expected to attend both the meeting and the lunch, and to proceed to a fund-raising event in Richmond, Va., on Friday.

It is unclear which version of Mr. Trump is likely to show up. After delivering a conventionally stage-managed victory speech on Tuesday, Mr. Trump returned to Twitter the next morning to attack the hosts of MSNBC’s “Morning Joe” for covering him unfavorably. Hours later, an aide to Mr. Trump called for barring a conservative radio host, Hugh Hewitt, from the Republican convention because of his criticism of Mr. Trump.

And even as his fund-raising efforts have intensified, Mr. Trump has resisted making phone calls to donors to solicit funds.

Edward F. Cox, the chairman of the New York Republican Party, said the party was grappling with unprecedented circumstances, brought about by the elevation of an untrained outsider unlike any presidential nominee in memory. Mr. Trump, he said, faced unusual pressure to build up his political operation and coordinate a message with Mr. Ryan.

Mr. Cox noted major opportunities coming up for Mr. Trump to reintroduce himself to voters and move on from his attacks on Mr. Curiel.

“You’re going to be looking to the naming of a vice-presidential candidate, you’re going to have an acceptance speech,” Mr. Cox said, “and this issue is going to be a very minor one.”