WASHINGTON—The Trump administration laid out ways to move more aggressively against Central Americans seeking asylum in the U.S., including stepping up pressure on Department of Homeland Security officials, amid a fresh setback in the courts and continued resistance on Capitol Hill.

A senior administration official on Tuesday said the White House is considering attempting to restrict the flow of remittances from the U.S., in order to discourage migrants. The White House is also considering giving parents a choice of being...