WASHINGTON — Senate Majority Leader Mitch McConnell demanded Thursday that House Democrats drop their insistence for changes in a $4.6 billion border aid package that the Senate passed with overwhelming bipartisan support and instead give final congressional approval to the legislation.

The Kentucky Republican spoke as House Speaker Nancy Pelosi, D-Calif., met with rank-and-file Democrats who had pushed a more liberal-leaning version of the legislation through their own chamber on Tuesday. Their measure has tougher care requirements for detained migrant families and children than the Senate package, seeks to handcuff President Donald Trump from diverting money to border security and has extra beds to detain more people.

“We’re not making predictions, threats, or anything else. We’re just saying this is a moral imperative for us to do the right thing for the children,” Pelosi said as she entered the meeting.

McConnell, R-Ky., took to the Senate floor to accuse House Democrats of trying to “dig back into that abolish-ICE playbook and throw a far-left partisan wrench” into efforts to approve the humanitarian aid bill. Some liberal House Democrats have said they favor abolishing the federal Immigration and Customs Enforcement agency.

“We already have a compromise,” McConnell said, citing the Senate bill, which passed Wednesday on a lopsided 84-8 vote. He called that measure “the only game in town,” adding, “It’s time to quit playing games, time to make a law.”

Suggesting a fig leaf, McConnell said the White House might support making some changes administratively — which have less than the force of law — to address some Democratic concerns.

Democratic demands include more protections for the children, such as setting medical and hygiene standards at facilities, and a requirement that any death of a minor be reported within 24 hours.

Pelosi was considering a House vote Thursday on the changes Democrats are seeking. But with Congress under pressure from a horrific photo of young drowned migrants and accounts of brutal conditions for detained children — and lawmakers eager to break for a July 4 recess — pressure was building on all sides to end a weekslong stalemate that has blocked the measure.

The White House said it opposes changes planned by the Democrats. The funding is urgently needed to prevent the humanitarian emergency on the U.S.-Mexico border from worsening. Money runs out in a matter of days.

On Wednesday, Pelosi called Trump before he departed for the G-20 Summit in Osaka, Japan, with an appeal to make changes. Trump seemed open, but it’s unclear if the Senate will accept any amendments without assurances from the White House that Trump will sign the measure into law. Both chambers are racing the clock to come to an agreement before leaving town for the recess.

Trump said passing the legislation was urgent. “We are moving along very well with a bipartisan bill in the Senate,” he said. “It’s very far along and I believe the House is also going to also be getting together with the Senate to get something done. It’s humanitarian aid. It’s very important.”

Both House and Senate measures contain more than $1 billion to shelter and feed migrants detained by the border patrol and almost $3 billion to care for unaccompanied migrant children who are turned over the Department of Health and Human Services. The Senate measure is not as strict in setting conditions on the delivery of funding to care for unaccompanied children and contains funding opposed by House Democrats for the Pentagon and to ease a payroll pinch at Immigration and Customs Enforcement.

Both House and Senate bills ensure funding could not be shifted to Trump’s border wall and would block information on sponsors of immigrant children from being used to deport them. Trump would be denied additional funding for Immigration and Customs Enforcement detention beds.

Lawmakers’ sense of urgency to provide humanitarian aid was amplified by recent reports of conditions in a windowless Border Patrol station in Clint, Texas, where more than 300 infants and children were being housed. Many were kept there for weeks and were caring for each other in conditions that included inadequate food, water and sanitation.

The Border Patrol reported apprehending nearly 133,000 people last month — including many Central American families — as monthly totals have begun topping 100,000 for the first time since 2007.