But Mr. Pence said the White House was bargaining in good faith. The administration has now provided Democrats details they were seeking about how any border security money would be spent, and officials have incorporated at least one Democratic idea, to allow migrant children seeking asylum to apply in their home countries rather than make a treacherous journey to the United States border.

“The question I have is: When are the Democrats going to start negotiating?” Mr. Pence said.

The administration’s credibility continues to suffer, as Democrats call out Mr. Trump for falsehoods about the crisis, such as his assertion that former presidents had told him privately that they should have built the wall. On Monday, former President Jimmy Carter joined the list of presidents who said they had never discussed a border wall with Mr. Trump.

With talks to end the shutdown at a standstill, Mr. Pence said the president had directed the Office of Management and Budget to take steps to “mitigate” its effects, including an order to the Internal Revenue Service to issue tax refunds. Under previous shutdown plans — and interpretations of federal law — the I.R.S. was prohibited from dispensing tax refunds when Congress had not approved money to fund the Treasury Department, as is the case now.

On Capitol Hill, Democrats tried to use leverage of their own to force Senator Mitch McConnell of Kentucky, the majority leader, to come to the table and pressure Mr. Trump. On Monday, Democrats said they would vote against advancing a package of bipartisan Middle East policy bills slated for consideration this week unless Republicans allowed a vote on bills to reopen shuttered federal departments already passed by the House — a decision that could scuttle its prospects if Democrats stick together.

Senate Democrats did not indicate whether they are ready to block other bills, but their position raised the prospect that a significant portion of the chamber’s work could halt until the Senate gets to vote to reopen the parts of the government now closed.

The House, under Ms. Pelosi, passed a package of bills on Thursday, the day Democrats took control of the chamber, to reopen the one-quarter of the government without funding Mr. Trump’s wall. But Mr. McConnell has refused to consider them, insisting that he will not bring up any legislation that does not have Mr. Trump’s explicit support.