“Locking up people who pose no threat to the community for ever-longer periods of time is not a solution to the problems at the border,” Ms. Lowey said in a statement, adding that the committee would review the request and work with the White House and the Senate “where possible.”

The new emergency request for border funds arrived on Capitol Hill before Congress has been able to reach an agreement on a much larger emergency package for people recovering from the onslaught of natural disasters last year. Democrats are reviewing the latest Republican offer, from Senator Richard C. Shelby of Alabama, the chairman of the Senate Appropriations Committee, that includes additional money for Puerto Rico’s recovery from hurricanes in 2017. (Mr. Trump told Democrats on Tuesday, “I’ll stay out of it; just get it done,” according to a congressional official familiar with the meeting.)

Homeland security officials have said for months that their facilities and resources have been pushed beyond capacity because of a surge in Central American families seeking asylum at the border.

Next week, officials from Immigration and Customs Enforcement will start conducting DNA tests on some migrants to determine if they are actually relatives, homeland security officials said on Wednesday. The pilot program, which would involve comparing DNA swabs of children and the adults they are traveling with, would be used to weed out fraudulent families. The testing is voluntary and requires the consent of the migrant.

Since April 18, more than 100 migrants have been suspected of fraud, officials said. Thousands of migrants are approaching the border a day.

Kevin McAleenan, the acting homeland security secretary, said on Tuesday that the processing facilities at the border were built for single adult males, but they now house more asylum-seeking families and unaccompanied children, who require a longer processing time.

On a call with reporters, senior administration officials underscored that the situation at the border was becoming more “dire” each day and noted that apprehensions were expected to surpass one million by the end of the year.