House Democrats blinked Sunday, with Speaker Nancy Pelosi announcing they will vote this week to grant at least part of President Trump’s emergency border funding to deal with the humanitarian crisis.

The move was made just a day after Mr. Trump, at Mrs. Pelosi’s behest, agreed to put off a round of deportations for two weeks. He said he wanted to give Congress space to tackle the border issue more comprehensively.

Mrs. Pelosi’s new bill does not appear to fit the bill.

She said it will grant Mr. Trump the emergency humanitarian money he has requested to provide better care for the migrant children and families who are overwhelming the border. But she said it will not support the rest of the president’s plans.

“This legislation provides urgently needed humanitarian assistance for families, including funding for food, shelter, clothing, medical care and legal assistance, and will relieve the horrific situation of overcrowding and help prevent additional deaths,” the California Democrat said.

Even as she agreed to give him some of the money he wants, she took a swipe at Mr. Trump, blaming him for a border crisis that Democrats, just a few months ago, doubted — and even called “manufactured.”

“The president’s failed policies have exacerbated the situation at the border, where vulnerable children endure inhumane conditions that threaten their health, well-being and sometimes, tragically, their lives,” Mrs. Pelosi said.

House Democrats are playing catch-up to the Senate, which cleared a $4.6 billion emergency spending through committee last week on a near-unanimous vote.

Led by Republicans, the Senate bill includes all of the money Mr. Trump requested to fund the Health Department’s Unaccompanied Alien Children (UAC) program, which is in danger of running out of money without an infusion.

The Senate bill also funds some of the Pentagon’s troop deployments and includes money for Customs and Border Protection and for more immigration judges — though it does not include money for bed space that the president wanted for Immigration and Customs Enforcement (ICE), the deportation agency.

Senators will give final approval to their bill in a floor vote this week, but it’s not clear whether they will be able to reach an agreement with the House on a final bill before they leave for an Independence Day vacation at the end of this week.

Mrs. Pelosi also made no overtures to work on the broader factors drawing the illegal immigrants to the U.S. — lax policies that require the government to quickly release children and families into the community, making their deportations nearly impossible.

Of those who arrived in 2017, 95% are still in the U.S., ICE says.

Mr. Trump had ordered a major deportation operation that was to have begun Sunday.

It was intended to track and arrest some of the illegal immigrant families who were part of a special pilot program that sped up their cases. Officials say 90% of the migrants didn’t bother to show for their hearings and were ordered deported in absentia.

ICE earlier this year sent notices to 2,000 of them, trying to arrange orderly deportations so officers didn’t have to track them down in the community, but ICE’s acting director said few took them up on the offer.

Democrats and immigrant rights activists protested the deportation plans nonetheless and insisted that the illegal immigrants be allowed to stay.

Mr. Trump, who Saturday morning said the deportations were necessary to send a signal, reversed himself Saturday afternoon at Mrs. Pelosi’s urging.

“At the request of Democrats, I have delayed the Illegal Immigration Removal Process (Deportation) for two weeks to see if the Democrats and Republicans can get together and work out a solution to the Asylum and Loophole problems at the Southern Border. If not, Deportations start!” the president posted on Twitter.

On Sunday, Democrats signaled they wouldn’t take him up on the offer.

“Democrats do not trust this president to implement a humane policy when it comes to the immigrants,” Rep. Adam Smith, Washington Democrat, said on CBS’s “Face the Nation” program.

Mr. Trump had called for action on the incentives spurring the surge of illegal immigrants to the U.S.: the court-ordered policy that requires families to be quickly released into communities, and the low bar for starting an asylum claim and thus earning the release. Most of those asylum claims are eventually rejected, but the migrants often refuse to leave.

The president calls those loopholes.

Democrats, though, call them important humanitarian protections, saying migrants must be allowed the chance to make their asylum cases even if they will be eventually rejected. Mrs. Pelosi called the illegal immigrant families “hardworking members of our communities and our country.”

Mr. Trump on Sunday was pessimistic that Democrats will cooperate.

“Probably won’t happen, but worth a try. Two weeks and big Deportation begins!” he tweeted.

Several news reports suggested another reason he postponed the immigration operation. They said the leak of the impending effort erased the element of surprise, giving immigrant rights activists time to coach illegal immigrants on how to avoid getting picked up.

The two-week window Mr. Trump set will close July 7, which is just before Congress is scheduled to return from its weeklong Independence Day vacation.

In the meantime, lawmakers will try to find agreement on the spending side as they eye the latest numbers detailing the extent of the problem.

More than 140,000 illegal immigrants were snared at the border in May alone — and the vast majority were families and children, setting an all-time record.

Under current policies, they must be quickly released. But they are coming in numbers so great that Border Patrol agents can’t process them quickly enough for release, fueling the overcrowding.

At one point, facilities built to hold perhaps 4,000 migrants along the southwestern border were holding 19,000. That number stood at about 15,000 last week, according to Homeland Security officials.

The Health and Human Services Department, meanwhile, has 13,700 children in its care and has fewer than 450 additional beds open — but the Department of Homeland Security is holding 1,400 at the border while they wait for a transfer.