Senators on Wednesday voted 84-8 in favour of the legislation, which includes $3 billion for humanitarian aid to improve conditions for undocumented migrants in detention, reports Efe news.
Though similar in many respects, the House bill omits extra money for the Defence Department as well as funds to pay overtime to Immigration and Customs Enforcement personnel.
The proposal also reinstates aid for El Salvador, Guatemala and Honduras, which President Donald Trump suspended because of the continuing northward flow of migrants from those countries.
Another provision of the House bill would require Customs and Border Protection to enact health and hygiene standards for individuals in custody - a response to recent revelations about squalid conditions at facilities where minors are being held.
The House draft was defeated 55-37 in the Senate and Trump has already threatened to veto the Democratic legislation.
The usual practice in these situations is for senators and representatives to meet in conference and reconcile the rival bills, but Republican and Democratic leaders indicated they had no plans to go that route.
Sen. John Thune of South Dakota, the second-ranking Republican in the Senate, said that going to conference wasn't a "viable" option.
"The House knows that they can't get a signature on their bill, and most of what they want is in our bill and ours is a bipartisan bill," he said.
Democrats, meanwhile, said they would only accept the Senate bill if Republicans agreed to make some changes.
"They passed their bill, we respect that. We passed our bill, we hope they will respect that," House Speaker Nancy Pelosi said later on Wednesday.
Given the distance between the two parties and the two chambers of Congress, the prospects for a bill reaching Trump's desk before the July 4th congressional recess appear dim.
More than 132,000 undocumented migrants were detained on the southern border in May, the highest one-month total since 2006.
( With inputs from IANS )