Vice President Mike Pence blasted congressional Democratic leaders Monday for refusing to negotiate with the administration on a border-security solution that would end the partial government shutdown.

“When are the Democrats going to start negotiating?” Mr. Pence asked in a session with reporters at the White House. “There is a crisis at the southern border. We just need the Democrats to start negotiating.”

Mr. Pence and other top administration officials held talks with Democratic congressional staffers over the weekend, but lawmakers won’t take part in the discussions until President Trump agrees to reopen the government. About 800,000 federal workers are not being paid during the shutdown.

Mr. Trump will give a prime-time address to the nation Tuesday night about his demands for funding for a border barrier, a proposal that Democrats reject outright.

“Our position is very simply this: there is a security and humanitarian crisis at the southern border,” Mr. Pence said. “Their position has been very clear. They refuse to negotiate until the federal government reopens.”

He said the president has invited Democratic leaders back to the White House this week for new talks, but so far there has been no response.

“We hope they take us up on it,” he said.

The vice president said Mr. Trump is still considering declaring a national emergency to fund the wall unilaterally. But he said Mr. Trump hasn’t made a decision on that option.

He also said that the White House is willing to consider emergency supplemental funding to cover the $5.7 billion request for a steel barrier, and more money for humanitarian aid and other needs at the border, such as more immigration judges.

Administration officials say there is general agreement with Democrats on providing an additional $800 million for urgent humanitarian needs at the border, including enhanced medical supplies and services. Two sick migrant children have died in the past month after crossing the border and being taken into custody by federal authorities, and about 50 illegal immigrants per day are in need of medical treatment at hospitals.

Mr. Pence said there is also agreement with Democrats on spending $675 million for “non-intrusive” inspection technology to search vehicles for drugs, weapons and other illegal materials.

But Democrats haven’t budged on the administration’s request for $5.7 billion for a barrier at the border, $563 million for 75 more immigration judges, or $211 million to hire another 750 Border Patrol agents.

Homeland Security Secretary Kirstjen Nielsen told reporters that the crisis at the border has grown significantly worse since last February, with about 2,000 illegal immigrants per day apprehended crossing the border.

“There’s a real sense of urgency. The crisis is getting worse,” she said.

Ms. Nielsen said the problem is compounded by a rise recently in the number of unaccompanied children and families in the overall population of illegal immigrants crossing the border.

White House officials said they are taking steps to minimize the problems for federal employees and others during the shutdown within the limits of the law.

Office of Management and Budget acting Director Russell Vought said the administration made sure that Coast Guard members received their pay, that flood insurance policies are renewed, and that the government is taking steps to ensure that trash is collected at national parks.

He also said the IRS will continue to process tax refunds during the shutdown.

“Tax refunds will go out,” Mr. Vought said.