The Trump administration is considering a revised version of its family separation tactic to cope with an influx of migrants at the U.S.-Mexico border that would force parents to choose whether to remain detained as a family or agree to a separation to keep their children out of custody, according to administration officials.

The administration weighed the new policy as President Trump insisted that he has no plans to separate families, falsely claimed that President Barack Obama carried out the same plan and maintained that his decision to halt the practice last year was the reason so many Central Americans have been coming to the United States.

“We’re not looking to do that now,” the president told reporters in the Oval Office, when asked to respond to reports that the White House is planning to separate families again. “But it brings a lot more people to the border when you don’t do it.”

Administration officials said Tuesday that while a return to the previous family separation tactic, known as “zero tolerance,” is not in the works, the White House is considering a “binary choice” policy, which would give parents the option of remaining in detention with their children or allowing their children to be separated and placed with another caregiver.

Trump’s comments came amid growing concerns among Senate Republicans about the shake-up at the Department of Homeland Security, with the ouster of Secretary Kirstjen Nielsen, the rescission of Ronald Vitiello’s nomination to be director of Immigration and Customs Enforcement and the departure of U.S. Secret Service Director Randolph D. “Tex” Alles. Alles and Vitiello reported to Nielsen.

Claire Grady, the DHS acting deputy secretary, was forced out Tuesday as well — her removal a necessary administrative step for U.S. Customs and Border Protection Commissioner Kevin McAleenan to take charge as acting secretary Wednesday.

The administration is struggling to deal with increasing numbers at the Mexican border. According to statistics released by Customs and Border Protection on Tuesday, more than 103,000 migrants were taken into custody along the border in March, the highest one-month total in more than a decade. They included nearly 9,000 unaccompanied minors and 58,000 migrants who arrived as part of family groups.

However, the president’s longtime allies on Capitol Hill expressed opposition to any return to a family separation policy, even “binary choice.” Sen. Ron Johnson (R-Wis.), chairman of the Senate Homeland Security and Governmental Affairs Committee, said he was working on bipartisan legislation addressing the border crisis and would be open to prohibiting the practice as part of that bill.

“If anybody in the administration is thinking about that, I’m opposed to it, and most members of Congress would be opposed to it . . . it’s simply not acceptable,” said Johnson, who said he shared his reservations about the “growing leadership void” at DHS with acting White House chief of staff Mick Mulvaney in a Monday night phone call.

Questioned specifically about “binary choice,” Johnson said, “That is not going to work.”

During Trump’s “zero tolerance” prosecution initiative last spring, border agents separated more than 2,700 children from their parents before the president was forced to reverse course in the face of public outrage.

But the number of migrant families at the border has reached record highs since Trump’s executive order last June halted the practice.

“I’m the one who stopped it,” Trump said, before repeating his discredited claim that family separations were an Obama-era practice. “President Obama had child separation.”

The department has no statistics on the number of children separated from parents under Obama, but current and former DHS officials acknowledge that it was rarely done, and typically only in cases when the child’s safety was considered at risk.

It was not until the Trump White House launched “zero tolerance” that U.S. agents began splitting parents and children in a systematic way, implementing a tactic that DHS officials under Obama considered too extreme.

Trump officials assert that “binary choice” would be legal because criminal defendants in the United States are often separated from their children when jailed. “That’s why we know it’s legal,” said one administration official, who spoke on the condition of anonymity to describe the plan.

“If you cannot separate children from their parents for purposes of imposing criminal sanctions on an adult, then you wouldn’t be able to prosecute anybody who has children, and we’d have to redo the entire criminal justice system,” the official said.

The overarching goal would be the same: keep adults in detention until their immigration claims are adjudicated by courts, to end the “catch and release” model Trump bemoans.

The implication is that adults who chose to keep their children with them in detention would waive protections granted under a decades-old court order that limits the number of days children can be detained.

Three administration officials said Tuesday that “binary choice” remained an option under discussion and that DHS had not begun making operations-level preparations to implement it.

Officials also said there were significant doubts within DHS over the feasibility of “binary choice” because Immigration and Customs Enforcement lacks the detention capacity to hold large numbers of families if parents opt to stay with their children.

McAleenan, whom Trump put in charge of DHS after removing Nielsen on Sunday, has not advocated a resumption of separations but has said authorities need the ability to hold families in custody while their asylum claims are processed.

He has proposed the expansion of family detention facilities that would offer a more kid-friendly setting than immigration jails, with recreational and educational programming, and better food and medical care.

Sen. Susan Collins (R-Maine) said she was alarmed by suggestions that the administration might institute “binary choice.”

“There’s no doubt that this flood of families over our borders is a legitimate concern and problem, but separating children from their parents, unless the parent is abusive or there is some other valid reason, is never a good idea,” Collins said.

In his comments to reporters Tuesday, Trump criticized the use of Border Patrol holding pens with chain-link partitions that have been widely derided as “cages.” Images of migrant children wrapped in plastic sheeting behind chain link exacerbated the backlash against Trump’s separations when they appeared last spring.

“Those cages that were shown — I think they were very inappropriate — were by President Obama’s administration, not by Trump,” the president said.

On that point, the president was correct: The facility with the chain-link pens is inside the Border Patrol’s Central Processing Center in McAllen, Tex., built in 2014.

The facility was established in a converted warehouse that the Obama administration opened to cope with record numbers of Central American minors arriving at the border, leaving Border Patrol stations dangerously overcrowded.

The open-floor facility uses the fencing to keep boys and girls apart, and to keep mothers with children in a separate area whose see-through barriers allow a small number of agents to supervise large numbers of minors. Migrants have nicknamed it “la perrera” — the dog kennel.

The Trump administration argues that Central Americans have been gaming the U.S. asylum system, taking advantage of laws that allow people to seek protection from persecution. Trump alleges that people are submitting false claims and that the U.S. immigration system is not tough enough on them. Those seeking asylum can be released into the United States while they await court hearings, which can be delayed months or years because of a massive backlog. U.S. courts have limited the government’s ability to hold families with children in detention.

“Now I’ll tell you something, once you don’t have it, that’s why you see many more people coming,” Trump said, referring to family separations. “They are coming like it’s a picnic, because, ‘Let’s go to Disneyland.’ ”

Trump also challenged claims that the spate of recent dismissals of top DHS leaders amounted to an attempt to clean house. He has expressed frustrations with the agency and its inability to reduce the number of migrants entering the country via the southern border.

“I never said I’m cleaning house,” Trump said. “I don’t know who came up with that expression. We have a lot of great people over there. We have bad laws.”

DHS officials say the White House’s moves have left the department “decapitated” and in administrative disarray, which has frustrated congressional Republicans. DHS general counsel John Mitnick is expected to be removed in the coming days. And a prominent GOP senator, Sen. Charles E. Grassley (R-Iowa), has expressed alarm to the White House about rumors that Lee Francis Cissna, the director of Citizenship and Immigration Services, will be forced out.

Trump was flooded last week with senators calling to urge him not to close the border, and this week senators have begun calling asking him to pick officials for top DHS jobs whom they can actually confirm, according to two senior administration officials.

“We can’t have this degree of vacancies and this degree of chaos. It’s a critical agency,” said Sen. Mitt Romney (R-Utah).

John Wagner contributed to this report.