Nevertheless, throughout the day on Monday, Nielsen and Attorney General Jeff Sessions defended the president's policy of splitting children from the adults who brought them into the U.S.
"We do not want to separate children from their parents, you can be sure of that," Sessions said. "If we build the wall, we pass some legislation, we close some loopholes, we won't face these terrible choices."
Nielsen said the separations are warranted in cases where the parents have broken U.S. law by entering the country illegally — and that her agency is treating them no differently from how the government treats parents who break any other laws. But the Trump administration is being accused of handling the cases in a way no other presidency has.
"This zero-tolerance policy is cruel. It is immoral. And it breaks my heart," former first lady Laura Bush said in remarks published Sunday night.
Bush wrote an opinion piece for The Washington Post in which she compared the DHS actions to the U.S. policy of incarcerating Japanese-Americans in camps during World War II.
First lady Melania Trump also weighed in over the weekend, with her communications director, Stephanie Grisham, sending a statement to CNN.
"Mrs. Trump hates to see children separated from their families and hopes both sides of the aisle can finally come together to achieve successful immigration reform," the statement read, concluding, "She believes we need to be a country that follows all laws, but also a country that governs with heart."
The DHS policy triggered a lawsuit in February; nearly a year earlier, members of the Trump administration had floated the idea of separating families as a potential deterrent.
A number of high-profile Trump officials have spoken in favor of the policy. Last month, Sessions said, "If you don't want your child to be separated, then don't bring them across the border illegally. It's not our fault that somebody does that."
Sessions spoke on Monday at the National Sheriffs' Association conference in New Orleans, where he received a lifetime achievement award. In his speech, the attorney general said the U.S. is having "an important conversation" over whether it will be "a country of laws" or "a country without borders."
One reason people voted for President Trump was "to end the lawlessness at our southern border," Sessions said, adding, "It's within our grasp. We can do it."
Sessions said that a loophole in U.S. policy had encouraged people to bring children to the U.S., under the belief that they would not be prosecuted. That resulted in a spike in illegal crossings, he said.
"We do not want to separate children from their parents," Sessions said. "We do not want adults to bring children into this country unlawfully, either, placing those children at risk."
Nielsen also spoke at the conference, before returning to Washington where she briefed reporters at the delayed White House briefing.
Describing how the DHS and Health and Human Services agencies handle those children, Sessions said, "They're not put in jail, of course – they're taken care of. They remain in the country, even though they don't have a lawful process to be here."
Sessions added, "That's an enormous cost that's being incurred by our government."
The backlash against the Trump administration's tactic grew into a protest march on Sunday, with hundreds of people heading to a tent city in the town of Tornillo, near El Paso, Texas, where children have been detained. The Father's Day march was organized by Rep. Beto O'Rourke, D-Texas, who is challenging Republican Sen. Ted Cruz. Cruz introduced a bill to prohibit family separations on Monday.
Overall, more than 10,000 children are currently in shelters run by the Office of Refugee Resettlement, which is part of the Department of Health and Human Services.
U.S. Customs and Border Protection has published a guide for families who have been separated after parents or guardians were charged with illegal entry, noting that children who are taken into government custody are transferred to the Office of Refugee Resettlement, "where your child will be held in a temporary child shelter or hosted by a foster family."
The guide sheet adds, "DHS and HHS can take steps to facilitate reunification with your child(ren)."
Last month, Nielsen told NPR that her agency is merely enforcing existing laws, and that criticism of the zero tolerance policy "is inappropriate and unacceptable."
That has not stopped what has now become a wave of criticism for the DHS policy of applying the law.
Religious leaders are speaking out against the policy, with Archbishop of Miami Thomas Wenski saying on NPR's Weekend Edition that the practice effectively "weaponizes" children.
Franklin Graham, son of the late evangelist Billy Graham and a prominent Trump supporter, told the Christian Broadcasting Network on Tuesday, "It's disgraceful and it's terrible to see families ripped apart, and I don't support that one bit."
In her tweets Sunday night, Nielsen accused others of distorting reality, saying, "This misreporting by Members, press & advocacy groups must stop."
The DHS secretary added, "if you are seeking asylum for your family, there is no reason to break the law and illegally cross between ports of entry."
The Obama administration also came under fire for its immigration and detention policies, with lawsuits over the practice of holding families in detention centers as well as the conditions at such facilities. By embracing a more hard-line policy, the Trump administration has set off speculation that it's using the threat of separating families as both a deterrent to immigrants and a political tool in the hopes of reshaping U.S. laws.
"You have many people in the Trump administration denying that they're using this as some sort of political leverage," NPR's Scott Detrow reports. "But President Trump is leaning into that pretty hard."
On Sunday, Trump tweeted, "The Democrats should get together with their Republican counterparts and work something out on Border Security & Safety. Don't wait until after the election because you are going to lose!"
On Monday, the president added, "Why don't the Democrats give us the votes to fix the world's worst immigration laws?"
In Congress, House Republicans have introduced a bill to end the practice of separating families, but it would also require Democrats to agree to federal funds for a border wall.
As that argument plays out, others are urging a new attempt to address the root causes of mass migration from people who want to flee troubled or impoverished conditions in countries such as El Salvador to Honduras.
Republican Rep. Will Hurd of Texas tells NPR's Steve Inskeep:
"There's different elements of the government that don't understand what's really going on. Kids are being separated from their parents. In the last two months, there's been about 2,000. The previous about year, it was almost 700. And 100 of those kids were under the age of 4."
Hurd added, "This is just absolutely unacceptable. Taking kids from their mothers is not preventing terrorists or drugs from coming into this country. And so, why we would even think that this is a tool that is needed to defend our borders is insane to me."
NPR's Danny Nett and Kelsey Snell contributed to this report.