“What the administration has decided to do is to separate children from their parents to try to send a message that, if you cross the border with children, your children are going to be ripped away from you,” she said on “Face the Nation” on CBS. “That is traumatizing to the children, who are innocent victims. And it is contrary to our values in this country.”

Former President Bill Clinton likewise spoke out, suggesting that Mr. Trump was using the widely denounced practice to leverage Democrats into accepting immigration limits in legislation they would otherwise oppose.

“These children should not be a negotiating tool,” he wrote on Twitter. “And reuniting them with their families would reaffirm America’s belief in & support for all parents who love their children.”

Hillary Clinton retweeted that message, adding, “YES!”

Contrary to the president’s public statements, no law requires families to necessarily be separated at the border. Attorney General Jeff Sessions’s “zero tolerance” announcement this spring that the government will prosecute all unlawful immigrants as criminals set up a situation in which children are removed when their parents are taken into federal custody.

Previous administrations made exceptions to such prosecutions for adults traveling with minor children, but the Trump administration has said it will not do so. While the president has blamed Democrats, his senior adviser, Stephen Miller, told The New York Times last week that it was “a simple decision by the administration to have a zero tolerance policy for illegal entry, period.”

But Kirstjen Nielsen, the secretary of homeland security, rejected responsibility for the separations in a series of tweets on Sunday. “We do not have a policy of separating families at the border,” she wrote. “Period.”

She distinguished between asylum seekers who try to enter the country at designated points of entry and those who arrive at other parts of the border. “For those seeking asylum at ports of entry, we have continued the policy from previous Administrations and will only separate if the child is in danger, there is no custodial relationship between ‘family’ members, or if the adult has broken a law,” she wrote.