She said Democrats were trying to advance “a bill that reflects more of our values,” but added: “We’re running out of time. We all saw what happened in Clint — there’s no time.”

Concern about the funding bill swelled over the weekend, after Mr. Trump tweeted on Sunday that he was suspending the raids for two weeks to provide time for a bipartisan compromise on changing asylum laws and closing immigration “loopholes.” His abrupt reversal came after Ms. Pelosi telephoned Mr. Trump to ask for a delay.

Ms. Pelosi praised the postponement, and in a strongly worded statement later on Sunday she called for passage of the emergency aid package, saying that it protects families and “does not fund the administration’s failed mass detention policy.” It would also do nothing to change asylum laws to meet Mr. Trump’s demands.

“As members of Congress and as Americans, we have a sacred moral responsibility to protect the human rights and the lives of vulnerable children and families,” she said. “To do anything less would be an outrageous and unacceptable violation of our oath and our morality.”

But even as the speaker was pressing to advance the bill, dozens of House Democrats were in revolt over it. In separate conference calls on Sunday, more than 30 members of the Progressive Caucus and more than 15 members of the Hispanic Caucus aired their concerns, many of them arguing that the legislation did not set high enough standards for migrant shelters or do enough to block money from going toward enforcement.

“We all want to address the problems at the border, but we don’t know that there are enough sticks in this bill to make sure that the Trump administration actually spends the money the way they’re supposed to,” said Representative Pramila Jayapal, Democrat of Washington and the co-chairwoman of the Progressive Caucus. “He’s creating these crises and then trying to point a finger at Democrats to give him more money, which he then uses for his own purposes.”

Ms. Jayapal said there was no reason to believe that the Trump administration would abide by any restrictions included in the legislation or standards dictated by the measure, given its “lawless” behavior when it came to immigrants.