“There’s a real sense that there’s a disconnect between the president and his staff on immigration issues, and people on all sides are seeking to exploit that disconnect,” said Alex Conant, a Republican strategist who advised Senator Marco Rubio of Florida, one of Mr. Trump’s rivals, in his 2016 bid for the White House. “This is what happens when you have a president who is not clear and consistent on what he will accept: It emboldens all parties to take positions that they won’t compromise.”
Senator Lindsey Graham, Republican of South Carolina, suggested that Mr. Trump was in the thrall of extremists on his staff pulling him back from more moderate instincts on immigration.
“His heart is right on this issue; I think he’s got a good understanding of what will sell, and every time we have a proposal, it is only yanked back by staff members,” Mr. Graham told reporters on Capitol Hill on Sunday. “As long as Stephen Miller is in charge of negotiating immigration, we are going nowhere. He’s been an outlier for years.”
Mr. Miller, 32, has been the ideological architect behind much of Mr. Trump’s immigration agenda and a tart-tongued and unapologetic true believer in the president’s “America First” approach to the issue. A former aide to Attorney General Jeff Sessions when he was in the Senate, he cut his teeth on Capitol Hill as a lonely gladiator against bipartisan efforts to overhaul the immigration system and provide a pathway to citizenship for roughly 11 million unauthorized immigrants.
The White House had a tart retort for Mr. Graham, a onetime opponent of Mr. Trump who in recent months seemed to be growing close to the president.
“As long as Senator Graham chooses to support legislation that sides with people in this country illegally and unlawfully instead of our own American citizens, we’re going nowhere,” said Hogan Gidley, a White House spokesman. “He’s been an outlier for years.”
The intraparty spat unfolded while Mr. Trump spent the weekend at the White House out of sight and off the airwaves, unusually disengaged, except for some phone calls, for a president who enjoys the limelight.
His only comment on the situation came on Twitter on Sunday morning, when he vented his frustration as the shutdown threatened to bleed into the workweek, complicating his plans for a trip on Wednesday to the World Economic Forum in Davos, Switzerland, and the run-up to his first State of the Union address on Jan. 30.
“If stalemate continues, Republicans should go to 51% (Nuclear Option) and vote on real, long term budget, no C.R.’s!” Mr. Trump said on Twitter, using the abbreviation for a continuing resolution, legislation to temporarily extend government funding.
He was referring to filibuster rules in the Senate, which effectively require a three-fifths vote, or 60 senators, to advance major legislation, rather than a simple majority. Republicans have 51 seats.
And he took a tone far different from the one he used this month in pitching a “bill of love” to address immigration, posting on Twitter that, “The Dems just want illegal immigrants to pour into our nation unchecked.”
Underscoring that hard-line position, his campaign released a TV advertisement featuring an undocumented man who killed two police officers, and saying Democrats who refused to support a government funding measure without progress toward an immigration deal were “complicit in every murder committed by illegal immigrants.”
Those who know the president best argue that leaving the legislative haggling to his staff is merely the style of an executive used to delegating the small stuff to his underlings.
“The misconception is that the president does not know what he does not know. In my experience, the reality is that the president knows what he does not know and does not think he needs to know it,” said Sam Nunberg, a former campaign adviser. “He’s a C.E.O. The tiny details are for his staff.”
But Mr. Trump is also a showman who is intensely focused on pleasing the audience in front of him at the moment, a habit that some confidants believe has led to misunderstandings about what the president is actually willing to accept in any deal. He often leaves people with the impression that he agrees with them, stressing whatever position is convenient at the time.
Mark Krikorian, the executive director of the Center for Immigration Studies, a group that presses for less immigration, said Mr. Trump had maintained his tough line on the issue despite occasionally talking about a compromise.
“He seems to make commitments that he is not going to keep,” Mr. Krikorian said. “His inclinations are hawkish on immigration, but he seems to like to be agreeable to people and nod his head when he’s at a meeting and people are saying things, and try to make a deal.”
Mr. Krikorian said that he did not subscribe to the “Svengali theory” of the White House that cast Mr. Miller as a puppet master on immigration, but that it often fell to him and Mr. Kelly to explain the nuances of certain terms or proposals to a president unfamiliar with them. The chief of staff alluded to that dynamic in a closed-door meeting with Democratic lawmakers last week and later in an interview with Fox News, enraging Mr. Trump.
Immigration advocates hold a darker view.
“The president should trust his instincts and cut a deal,” said Kevin Appleby, the senior director of international migration policy at the Center for Migration Studies. “He is president and should not be the puppet of a few immigration restrictionist staffers, including his chief of staff. The perception is that they have total control over him, to the detriment of the nation.”
Mr. Kelly, a retired four-star general who headed the United States Southern Command and was Mr. Trump’s first homeland security secretary, has emphasized immigration enforcement inside the country rather than policing the borders while Mr. Trump has indicated that is not as high a priority for him.
On Sunday, Mr. Kelly fielded most of the calls from Senator Mitch McConnell, Republican of Kentucky and the majority leader, and the House speaker, Paul D. Ryan of Wisconsin. The president was urged for a second day to step back from the fray, and for a second day he vented to aides that he wanted to do more to get involved.
Yet when Mr. Trump has become engaged, he has sometimes created problems for himself and his party.
Mr. Trump has demonstrated confusion over time about the details of immigration policy, including during a televised meeting in the Cabinet Room this month with lawmakers of both parties.
When Senator Dianne Feinstein, Democrat of California, said she wanted a “clean DACA bill,” Mr. Trump quickly agreed, only to have Representative Kevin McCarthy of California, the majority leader, pipe up to explain that meant accepting a stand-alone bill to legalize a group of undocumented immigrants brought to the United States as children, without any security measures or other conditions the president had cited as priorities.
During a closed portion of that meeting, Mr. Trump snapped at staff members for handing out a sheet of paper he had not seen before that included an elaborate plan for border security.
“The president looked at it and said: ‘Who did this? This is way too much. I didn’t approve this,’” Mr. Graham said on Sunday.
At that same session, he added, Mr. Trump had talked about a request of $18 billion for border security, and said he could build a wall for less.
“So what does the White House staff do a couple days later? They pitch a proposal for $33 billion,” Mr. Graham said. “That’s just not credible.”