WASHINGTON — House Democrats spurned an invitation from President Trump to have a bipartisan lunch at the White House on Tuesday, as increasingly agitated lawmakers in both parties and on both sides of the Capitol began casting about for a compromise to end the standoff that has pressed the partial government shutdown into its 25th day.

The Democrats’ absence from the lunch was the latest indication that the party is continuing to stand firm against Mr. Trump’s demand that any proposal to reopen the government must include $5.7 billion for a wall on the southwestern border. White House officials, who had issued the invitation in the hopes of showing fissures among Democrats, used the snub to deflect responsibility for the prolonged shutdown, arguing that Mr. Trump was the one trying to end the impasse. A slew of recent polls have found that the public largely blames Mr. Trump and Republicans for the continuing dysfunction.

“Unfortunately, no Democrats will attend,” Sarah Huckabee Sanders, the White House press secretary, said in a statement about the lunch, which was to include nine rank-and-file House Republicans. “The president looks forward to having a working lunch with House Republicans to solve the border crisis and reopen the government. It’s time for the Democrats to come to the table and make a deal.”

Speaker Nancy Pelosi of California and Senator Chuck Schumer of New York, the Democratic leader, have refused to negotiate over border security until the government reopens, but Mr. Trump has ruled out separating the two issues. While the Democrat-led House has passed several bills to end the shutdown, Senator Mitch McConnell of Kentucky, the majority leader, has said he will not advance the legislation knowing that the president will not sign it.

Behind the scenes, Republicans and Democrats on Capitol Hill toiled without success to find a solution Mr. Trump would accept.

In the House, some freshman Democrats who won in districts carried by Mr. Trump in 2016 planned to meet on Tuesday to talk about whether — and how hard — to push their leaders to negotiate with the White House.

“Maybe it’s an outlier view compared to some others in the Democratic Party, but I believe we have a responsibility to get in a room and negotiate,” Representative Elissa Slotkin, Democrat of Michigan, said in an interview.

Ms. Slotkin said she convened the session with other newly elected Democrats because she and her colleagues wanted “a better sense of what our appropriate voice should be” on the matter. She added that she was trying to assist Ms. Pelosi, not undercut her.

But Ms. Slotkin said she got an earful from constituents over the weekend, and promptly called the speaker’s office on Monday to report what she had heard. She has also been talking quietly with Republicans — she would not name them — about how to find a way out of the stalemate.

Yet Democratic leaders expressed confidence that their members would not break from their leaders’ strategy, notwithstanding Mr. Trump’s attempts to entice them.

“Is anybody surprised that the president is trying to get votes anywhere he can get votes?” Representative Steny H. Hoyer of Maryland, the majority leader, told reporters on Tuesday. “We are totally united — totally.”

In a closed-door meeting of House Democratic leaders on Monday, Ms. Pelosi and Mr. Hoyer said they had no problem with the idea of rank-and-file Democrats meeting with Mr. Trump. Ms. Pelosi said such a session would give lawmakers a sense of “what we’ve been dealing with” in a series of tense meetings with the president in the Situation Room since the shutdown began. “They’ll want to make a citizen’s arrest,” she added at one point, according to an official in the room who spoke on condition of anonymity because the conversation was private.

In the Senate, a group of Republican and Democratic senators met privately on Monday in the office of Senator Joe Manchin III, Democrat of West Virginia, to try to find a way out of the gridlock, but the talks yielded no breakthrough.

The group included Senator Lindsey Graham, Republican of South Carolina, who tried last week to forge a compromise that would pair border security funding with legal status for certain groups of immigrants facing deportation. Democrats in the room said no progress could be made on such a deal until the government was reopened, two officials familiar with the talks said, and Republicans agreed.

“What I would hope is that the president would reconsider” and allow the government to reopen for a brief period while senators worked on a compromise, said Senator Lamar Alexander, Republican of Tennessee, who attended the session.

But the president has repeatedly ruled out doing so, and privately told Democrats this month that such a move would make him look foolish.

As the gridlock continued, the Trump administration was searching for ways to lessen the pain of the partial shutdown for those affected. Kirstjen Nielsen, the secretary of homeland security, said she was working on legislation to ensure that the Coast Guard, the only branch of the military going without pay during the lapse in funding, would be compensated.

The acting director of the Office of Personnel Management has been working with payroll providers to ensure that some federal workers going without compensation would receive back pay within a few days of when the government reopens, according to a senior administration official. The official, who spoke on condition of anonymity without authorization to discuss the plan, said it would affect more than half of the federal work force that is not getting paid.

Sheryl Gay Stolberg and Annie Karni contributed reporting.

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