But Democrats said they were left with the impression that the most senior officials in attendance were arguing that they were not empowered to make any agreements. Mr. Pence noted that no numbers for wall funding would be discussed because the meeting was not a principals’ meeting. And when he introduced the president’s son-in-law, Jared Kushner, as the person who knew Mr. Trump better than anyone, Mr. Kushner demurred, saying that distinction belonged to his wife, Ivanka Trump, the president’s eldest daughter, who was not in the meeting.

A White House staffer in attendance described the weekend discussions as productive and cordial, but blamed Ms. Pelosi for being unwilling to look like she was negotiating with Mr. Trump to end the shutdown.

While there was little common ground, some small advances were made, White House officials said. Republicans agreed that a concrete wall was a non-starter, and that any text of a bill would refer to steel and not concrete.

And even though Mr. Pence said numbers would not be part of the weekend discussions, the secretary of the Department of Homeland Security, Kirstjen Nielsen, on Sunday provided Democrats with a written statement outlining why the administration needed $5.7 billion for a barrier along the southern border.

In the meeting, Ms. Nielsen also gave the full briefing she had prepared to give Democratic leaders last week in the Situation Room, outlining what the White House has described as a crisis at the border. Several Democrats in the room, according to the White House official, said they agreed with her that a humanitarian border crisis did exist. But the Democrats who were interviewed said the two sides remained far apart about what constituted the correct response to a humanitarian crisis.

Stephen Miller, the White House policy aide who has been the architect of the president’s immigration policies and its public face, occupied a seat along the edge of the room at Saturday’s meeting, not at the table, and spoke infrequently, according to attendees. He interjected memorably only once, to argue that the Flores agreement, which limits how long the government can detain immigrant children to 20 days, was creating a loophole for smugglers.