The proposal, which had been called a "compromise" bill by Republicans, would have provided some respite to youth brought to the country illegally as minors, as well as some $25 million in funding for part of a wall along the border with Mexico.
The European crisis mirrors the one in the United States, where a broad-ranging Republican immigration bill was set for a vote Wednesday, with little certainty that it would survive.
In a 5-4 decision June 26, the U.S. Supreme Court upheld President Donald Trump's travel ban on people entering the U.S. from some Muslim-majority countries, saying the president's action was within his power.
Hundreds of American adults have called the Washington headquarters of the U.S. Conference of Catholic Bishops seeking to provide foster care for the separated children.