Sen. Lindsey Graham, chairman of the Judiciary Committee, said there will be a first vote in the Appropriations Committee next week.
“I think it’ll be money well spent,” Mr. Graham said.
A majority of the $4.5 billion Mr. Trump has requested would go to to the federal Health Department, which is running out of money to house and care for the unaccompanied illegal immigrant children. Other money would go to replenish accounts at Customs and Border Protection, where facilities designed to hold perhaps 5,000 people now hold 19,000 migrants.
But the bill presents a dilemma for Democrats, who are reluctant to give Mr. Trump a victory on anything related to his border plans, saying they don’t want to reward him after his fight over money to build the border wall earlier this year or after his zero tolerance policy that led to family separations.