Congress is trying to rush $4.5 billion in emergency humanitarian aid to the southwestern border while placing new restrictions on President Trump’s immigration crackdown, spurred on by disturbing images of suffering migrant families and of children living in squalor in overcrowded detention facilities.

But with a House vote on the package planned for Tuesday, some Democrats are revolting over the measure, fearing that the aid will be used to carry out Mr. Trump’s aggressive tactics, including deportation raids that he has promised will begin within two weeks. Republicans are siding with the White House, which on Monday threatened a veto. They oppose restrictions in the measure that are meant to dictate better standards for facilities that hold migrant children and to bar the money from being used for enforcing immigration law.

Those twin challenges have left the fate of the bill up in the air, even as evidence of deplorable conditions at the border underscores both the urgent need for the money and the bitter rift over Mr. Trump’s policies.