The negotiations on Saturday focused on priorities for security rather than a dollar figure for the border wall, the vice president’s office said. While Mr. Trump has stood by his $5.7 billion demand, Senate Democrats have offered $1.3 billion for border security, including fencing, while Speaker Nancy Pelosi and the top Senate Democrat, Chuck Schumer of New York, have repeatedly said that they will not agree to any wall funding. Ms. Pelosi has called a border wall an “immorality.”

The vice president’s office said that Mr. Pence had reiterated the president’s position that any deal needed to include funding for the wall. The office also said that Democrats had requested additional information from the Department of Homeland Security about its needs to deal with border issues.

Democratic staff members asked for a formal budget justification for the administration’s insistence on its $5.7 billion proposal, a Democratic official familiar with the discussion said, adding that Mr. Pence made clear that the White House would not budge from that figure. The Democrats told the vice president that there would be no movement on the dollar figure until after the government is reopened.

It is unclear just what kind of authority Mr. Trump has granted Mr. Pence to speak for him in negotiations. Last month, when Mr. Pence made a $2.6 billion counteroffer to Democrats in an effort to avert the shutdown, Mr. Trump quickly shot down the proposal.

During the talks on Saturday, Kirstjen Nielsen, the homeland security secretary, offered a briefing on what the administration has deemed a “crisis” at the border. Ms. Nielsen had tried to give a similar briefing earlier in the week to congressional leaders and White House officials gathered in the Situation Room, but she was cut off by Ms. Pelosi, who questioned Ms. Nielsen’s facts.