WASHINGTON — The television is on. The phone is never far away. And President Trump is repeatedly calling allies such as members of Congress and conservative radio hosts, telling them privately that he will not give in on his demand for funding for a border wall.

What the president who campaigned on his ability to cut deals has not done, nine days into a partial government shutdown over his signature campaign issue, is reach out to Democratic congressional leaders to strike one.

Virtually alone in the West Wing since the shutdown began, Mr. Trump has instead taken to Twitter to excoriate Democrats, and highlight that he canceled his own vacation to his private club in Florida while lawmakers left the city. He has lamented the negativity of the news media coverage, which has included repeated airings of Mr. Trump’s declaration in the Oval Office that he would not blame Democrats for a shutdown, according to people familiar with his thinking.

Even as some lawmakers floated compromises on Sunday, Democrats prepared to pass a bill to fund the government as soon as they take control of the House on Thursday. Like the Democrats, Mr. Trump appears to have dug in. And the uncertainty over what he might sign threatens to indefinitely drag out a shutdown that has affected 800,000 federal workers and shuttered parts of nine cabinet-level departments.

After Senator Lindsey Graham, Republican of South Carolina, met with Mr. Trump over lunch on Sunday, he said the president would not accept any deal without funding for the wall. But he remained optimistic that a compromise could be reached and encouraged both sides to come together.

“At the end of the day, there’s a deal to be had,” he said on Sunday. “We need to start talking again.”

Still, Mr. Graham said after the meeting that the president had not signed on to his potential compromise, which would provide wall funding in return for work permits for the young undocumented immigrants known as Dreamers. Democrats also have no interest in such a plan right now.

And there were other signs of a lengthy shutdown fight from the White House: The president has a new acting chief of staff, Mick Mulvaney, who is not averse to government shutdowns, and some advisers see the timing of the fight as preferable to a year from now, when Democrats will be preparing for the first votes in the 2020 presidential primaries.

The president is also concerned that if he makes a deal, his core political support will falter, and his voters will see him as inauthentic after he talked about a wall in rally after rally for three years. As Mr. Trump was poised to sign a compromise bill that would have funded the government through February, conservative commentators berated him as “gutless,” and some hard-line House Republicans urged him to reconsider. He backed away from the bill soon after.

Senator Lindsey Graham, Republican of South Carolina, expressed optimism for a deal after meeting with Mr. Trump on Sunday, but said the president had not signed on to his potential compromise.

Senator Lindsey Graham, Republican of South Carolina, expressed optimism for a deal after meeting with Mr. Trump on Sunday, but said the president had not signed on to his potential compromise.CreditPablo Martinez Monsivais/Associated Press
Adding to the mixed messages over what eventual deal the president would accept, John F. Kelly, the outgoing White House chief of staff, said in an interview published Sunday that the administration had backed away from the idea of a solid concrete wall long ago, even though Mr. Trump had dangled “steel slats” as a potential enticement to lawmakers only in recent weeks.

How it all ends is a mystery.

The president’s counselor, Kellyanne Conway, and personal lawyer, Rudolph W. Giuliani, faulted the presumptive incoming House speaker, Nancy Pelosi of California, for leaving for Hawaii while the government was partly shuttered. They blamed Democrats for not reaching out to forge a compromise.

“Those who have completely walked away from the table are doing no justice to the people of this country,” Ms. Conway said on “State of the Union” on Sunday morning.

People close to Mr. Trump consider his statement about owning a shutdown regrettable. But White House officials and others close to the president also believe that a shutdown is unlikely to help Ms. Pelosi as she takes over as part of the new House majority, and that Democratic officials risk looking unreasonable.

Pelosi allies say that they think the shutdown will help her look like the adult in the room, because she is going to quickly move to end it, demonstrating that Democrats are serious about governing.

Ms. Conway did not rule out a presidential veto of the bill the House Democrats plan to advance.

“It depends what’s in it,” she said.

Senator Richard C. Shelby, the Alabama Republican who is chairman of the Appropriations Committee, said the Senate would not take up the House bill unless Mr. Trump backed it. Sounding frustrated, Mr. Shelby used an appearance on CBS’s “Face the Nation” to appeal to the president and Democrats to quit pointing fingers at one another.

“Whether it’s the president tweeting and blaming somebody or blaming the Democrats or whether it’s the Democrats blaming the president, it’s brought us to the impasse that we are today,” he said.

“Nobody’s going to win this kind of game,” Mr. Shelby said. “Nobody wins in a shutdown. We all lose and we kind of look silly.”

Before leaving for the Christmas break, House Republicans pushed through a bill that included $5.7 billion for wall funding and border security, knowing such a measure would be dead on arrival in the Senate, where Republicans need Democratic votes to pass spending legislation.

Vice President Mike Pence and Mr. Mulvaney subsequently met with Senator Chuck Schumer of New York, the Democratic leader, and offered a compromise: $2.5 billion for border security, including new fencing. Mr. Schumer countered with three different Democratic proposals, each of which would include $1.3 billion for fencing and border security.

Senator Richard Shelby, Republican of Alabama and the chairman of the Appropriations Committee, urged the president and Democrats to quit pointing fingers at one another.

Senator Richard Shelby, Republican of Alabama and the chairman of the Appropriations Committee, urged the president and Democrats to quit pointing fingers at one another.CreditErin Schaff for The New York Times
Now, negotiations are at a standstill, with each side insisting the ball is in the other’s court. Ms. Conway on Sunday accused Democrats of failing to negotiate with Mr. Trump, telling CNN: “They know where he is. He’s exactly where he’s been the entire time, working in Washington, D.C., in the White House.”

But a spokesman for Mr. Schumer complained that the White House was sending mixed signals, noting that Mr. Trump had yet to publicly embrace the Pence-Mulvaney offer.

“At this point, it’s clear the White House doesn’t know what they want when it comes to border security,” the spokesman, Justin Goodman, said Sunday. “While one White House official says they’re willing to compromise, another says the president is holding firm at no less than $5 billion for the wall. Meanwhile, the president tweets, blaming everyone but himself.”

The partial shutdown came about when Congress failed to pass seven spending bills, including for the Department of Homeland Security. While House Democrats are considering various options, they appear to be leaning toward funding Homeland Security through a so-called continuing resolution, which would keep spending at current levels, likely through February. The other agencies would be funded through separate measures.

Meanwhile, Mr. Graham floated his proposal, which would marry $5 billion for the wall with immigration law changes that might appeal to Democrats, including three-year, renewable work permits for the Dreamers, who were brought to the country illegally as children. Their status has been uncertain since Mr. Trump moved to end a program that protected them known as Deferred Action for Childhood Arrivals, or DACA.

But Mr. Graham has made similar proposals in the past, and they have gone nowhere on Capitol Hill. And Democrats have made clear that such a deal, which many DACA recipients do not want, is no longer of interest to them.

That is especially true with the potential for a case challenging Mr. Trump’s ending of the DACA program to make its way to the Supreme Court next year.

The budget fight is playing out amid the controversy over the deaths of two migrant children who were in the custody of Customs and Border Protection officials.

On Saturday, Mr. Trump tweeted that the children’s deaths were the fault of Democrats and their immigration policies, in his first public comments since the 7-year-old girl and 8-year-old boy died this month. Mr. Trump expressed no remorse for their deaths, and falsely stated that the parents of the children had said they were sick when they came in to the country.

In an interview with ABC News’s “This Week” on Sunday, the commissioner of Customs and Border Protection, Kevin McAleenan, said officials “did everything they could, as soon as these children manifested symptoms of illness, to save their lives.”

He said that the number of families with children crossing the border had risen, and that the system was not set up for children. “We don’t want them in Border Patrol stations,” Mr. McAleenan said. “We want them in a better scenario for these vulnerable populations that we are seeing.”