President Barack Obama said Friday the U.S. is considering a limited refugee program in Central America, for the first time allowing young people to apply for entry without first making the dangerous trek north.

With Congress deadlocked on a broader White House request for border funding, the refugee plan is one step the administration says it can take on its own to try to stem a surge in unaccompanied Central American minors illegally entering the U.S. from Mexico. White House officials said the program would be based in...