The president’s decision to target Mr. Obama’s program for young illegal immigrants, known as Deferred Action for Childhood Arrivals or DACA, will open the fall on a divisive note. But his advisers hope that delaying the effect by six months will force Congress to step up and put the program on a firm constitutional foundation.

While Republicans feel burned by Mr. Trump’s attacks on Mr. McConnell and some of their colleagues, some said they have little choice but to find a way to come together.

“We have to work with the president,” Senator Roy Blunt of Missouri, vice chairman of the Senate Republican Conference, said on “Meet the Press” on NBC last weekend. “I think it’s a mistake to get in a fight with the president. It’s not a mistake to disagree when you disagree; it is a mistake to suggest that somehow this president, who was elected just as the Constitution prescribed, and has the responsibility to lead the country, that somehow we need to not work with this president.”

On Tuesday afternoon, Mr. Trump will host a group that has been dubbed the Big Six to discuss his tax overhaul — Mr. McConnell; House Speaker Paul D. Ryan; Treasury Secretary Steven Mnuchin; Gary D. Cohn, the president’s national economics adviser; and Senator Orrin G. Hatch of Utah and Representative Kevin Brady of Texas, the Republican chairmen of the tax-writing committees. On Wednesday, Mr. Trump will host another White House meeting with Mr. McConnell, Mr. Ryan and their Democratic counterparts, Senator Chuck Schumer of New York and Representative Nancy Pelosi of California.

Mr. Trump has long boasted of his art-of-the-deal negotiating skills and part of his appeal in last year’s election was the hope that he could use them to finally break through a paralyzed capital. “Deals are my art form,” he once said on Twitter. “Other people paint beautifully or write poetry. I like making deals, preferably big deals.”

His first seven months in office have yet to produce any big deals, a failure highlighted most notably by the collapse of an effort to repeal Mr. Obama’s health care program and replace it with a Republican-authored version. Allies said Mr. Trump’s approach to negotiations, however, is to hold out for the best deal possible until the last moment, so it is too soon to judge.

“My style of deal-making is quite simple and straightforward,” he once wrote. “I aim very high, and then I just keep pushing and pushing and pushing to get what I’m after. Sometimes I settle for less than I sought, but in most cases I still end up with what I want.”