WASHINGTON — The Homeland Security agency responsible for removing immigrants who are in the country illegally will run out of money by mid-August unless Congress approves President Barack Obama’s emergency request for $3.7 billion to help deal with a flood of child immigrants crossing the border illegally without their parents, Homeland Security Secretary Jeh Johnson says.

Additionally, Customs and Border Protection, whose 20,000-plus Border Patrol agents are responsible for arresting illegal border crossers, will be out of money by mid-September at the “current burn rate,” Johnson told the Senate Appropriations Committee on Thursday.

Johnson said if Immigration and Customs Enforcement and the Border Patrol run out of money, the Homeland Security Department “would need to divert significant funds from other critical programs just to maintain operations.”

While Johnson, Health and Human Services Secretary Sylvia Burwell and other administration officials made their pitch for the extra money, outlines of a possible compromise to more quickly deport minors arriving from Central America emerged Thursday. More than 57,000 child immigrants, mostly from Central America, have been caught crossing the Mexican border since Oct. 1.

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