“Unfortunately, no Democrats will attend,” Sarah Huckabee Sanders, the White House press secretary, said in a statement about the lunch. She added, “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 Democratic-led House has passed several bills to end the shutdown, Mr. McConnell has said he will not advance the legislation knowing that the president will not sign it.

“In a situation like this, where the president, in my view, is in the right place, trying to get the right outcome, as all of us have expressed, with regard to border security, of course not,” Mr. McConnell said.

Ms. Slotkin would not say whether she agreed with Ms. Pelosi’s stance that no negotiation was possible until the government reopened, citing a confidentiality pledge she said members of the group had taken about their discussion. But she said she and her colleagues were walking around with copies of legislation co-written by Representative Will Hurd, Republican of Texas, and Representative Pete Aguilar, Democrat of California, that drew bipartisan support last year. The measure paired border security improvements with legal status for some undocumented immigrants brought to the United States illegally as children, sometimes called “Dreamers.”

“There’s a lot of people who believe it’s important to work across the aisle, that we need a practical way forward, and they believe in figuring out practical solutions,” Ms. Slotkin said. “We know we’re not in charge, but we want to get caught trying to help.”

She said she had called Ms. Pelosi on Monday to relay complaints by her constituents over the weekend about the shutdown, and has also been talking quietly with Republicans — whom she would not name — 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.