WASHINGTON — Outlines of a possible compromise that would more quickly deport minors arriving from Central America emerged Thursday as part of President Obama’s $3.7 billion emergency request to address the immigration crisis on the nation’s southern border.

Republicans demanded speedier deportations, which the White House initially had supported but left out of its proposal after complaints from immigrant advocates and some Democrats. On Thursday, the top House and Senate Democrats pointedly left the door open to them.

“It’s not a deal-breaker,” said House Minority Leader Nancy Pelosi, D-Calif. “Let them have their face-saver. But let us have the resources to do what we have to do.”

Her spokesman Drew Hammill later clarified that any changes “must ensure due process for these children.”

In the Senate, Majority Leader Harry Reid, D-Nev., said: “I’m not going to block anything. Let’s see what comes to the floor.”

But opposition arose late in the day from a key Democratic senator, suggesting battles ahead before any deal could be struck.

“I can assure you that I will fight tooth and nail changes in the Trafficking Victims Protection Act,” Senate Judiciary Chairman Patrick Leahy said at a hearing on the situation, referring to the law Republicans want to change.

Noting that the arriving migrants include young girls trying to escape sex violence and gangs, Leahy said: “I’m not sure Americans all really feel we should immediately send them back.”

Reid and Pelosi made their comments as House Speaker John Boehner, R-Ohio, and Senate Republican leader Mitch McConnell, R-Ky., both said they didn’t want to give Obama a “blank check” to deal with the crisis of tens of thousands of unaccompanied children arriving at the Texas border, many fleeing gangs and drawn by rumors they would be able to stay in the United States.

The developments came as Obama’s Homeland Security secretary, Jeh Johnson, defended the emergency spending request at a hearing of the Senate Appropriations Committee.

Johnson told senators that without the money, the U.S. Immigration and Customs Enforcement and Customs and Border Patrol agencies would both run out of money in the next two months.