Sen. John Cornyn pondered Monday when asked whether anyone could win Senate confirmation as the new Homeland Security secretary.
The Texan came up with one name: Jeh Johnson, who was secretary for the last three years under President Barack Obama, overseeing the last border surge in 2014 and 2015.
Mr. Cornyn, the Senate’s second-ranking Republican, quickly told reporters he was joking about Mr. Johnson, but his point was clear. Getting any pick through the Senate is going to be rough for President Trump, who over the weekend ousted Secretary Kirstjen Nielsen.
Democrats, who for months had demanded Ms. Nielsen be fired, were torn. Some said they were happy to see her go, but others now said she should have stayed, if only to prevent chaos at her old department, and to prevent Mr. Trump from naming someone even “tougher” on immigration.
Mr. Trump has been on a housecleaning tear.
In addition to Ms. Nielsen, he withdrew the nomination of Ronald D. Vitiello to be director of U.S. Immigration and Customs Enforcement last week. Then on Monday the head of the Secret Service, Randolph D. “Tex” Alles, announced he will step down next month.
The department already lacked a full confirmed deputy secretary, emergency management director, inspector general and two undersecretaries — one for policy, and another for science and technology.
“Unfortunately, in an effort to fulfill his campaign promises, President Trump is trying to remake DHS into his own personal anti-immigration agency. That’s not good for the country and it’s certainly not good for a department that employs 240,000 federal workers who work to keep our country safe,” said Sen. Dianne Feinstein of California, the ranking Democrat on the Senate Judiciary Committee, which shares jurisdiction over the department.
She said even Republicans “may balk” if Mr. Trump tries to find a “tougher” nominee to lead the department.
The president’s pick to be acting secretary is Kevin McAleenan, currently the head of Customs and Border Protection.
But a top Democrat said Monday that pick is illegal. House Homeland Security Chairman Bennie G. Thompson said under the law, the order of succession would make the undersecretary for management the acting secretary. That position is currently held by Claire M. Grady, who is also serving as acting deputy secretary.
“I strongly urge you to follow the law in this matter,” Mr. Thompson wrote in a letter to Mr. Trump, adding he wanted to see “a suitable candidate” for the secretary’s job nominated as soon as possible.
He didn’t suggest any names, but the Venn diagram of someone Democrats think is suitable and that Mr. Trump would nominate is tough to picture.
While Mr. McAleenan hasn’t been nominated, his name has been floated as a prominent option.
Mr. Cornyn said he’s done a “tremendous job” leading the border agency, but wasn’t sure he’d have any luck at the top of the department.
“I’m not sure he’s going to have any more success than Secretary Nielsen in solving the underlying problem,” the senator said.
That problem is a set of U.S. laws that serve as enticement for illegal immigrants to make the trip north from Central America, convinced they can game the system to gain a foothold in the U.S. For the most part, they have been correct.
Mr. Trump and Democrats have clashed over whether the solution lies here in the U.S., with a change in the laws, or in Central America, where Democrats want to see a round of taxpayer-funded nation-building to improve conditions and cut down on the “push” factors.
Democrats also said Mr. Trump’s next nominee must be prepared to refuse to listen to the president on some key issues.
That Mr. Johnson’s name comes up, even in jest, from Republicans is also a testament to the former Obama Cabinet official, who has become a singular voice in recent weeks. Having overseen — and conquered — a similar migrant surge during his tenure, he has labeled the new, larger surge facing Mr. Trump a crisis, and urged careful action.
One major flashpoint for whomever Mr. Trump picks will be whether a new nominee would renounce last year’s zero-tolerance policy, which led to thousands of illegal immigrant children being separated from parents who were put into the criminal justice system.
“I will not support a nominee who does not forcefully and unequivocally denounce this administration’s policy of separating families at the border,” said Sen. Kamala D. Harris, California Democrat. “Cruelty is not a substitute for a smart, lawful and humane immigration policy.”
One immigrant-rights group said it doesn’t matter whom the president picks.
“Trump’s real DHS secretary has always been Stephen Miller who continues to order the government to make his and Donald Trump’s racist fantasies become real and deadly,” said Sanaa Abrar, advocacy director for United We Dream.
Several reports have said Mr. Miller, a senior policy adviser to the president, was the voice urging Mr. Trump to withdraw the nomination of Mr. Vitiello and to move on from Ms. Nielsen. He also apparently wants to see the ouster of U.S. Citizenship and Immigration Services Director Francis Cissna.
Democrats who for months had complained about Ms. Nielsen turned their fire on Mr. Miller on Monday.
“Donald Trump and Stephen Miller are creating massive gaps in our Homeland Security leadership,” said Rep. Lucille Roybal-Allard, California Democrat and chairwoman of the House subcommittee that handles Homeland Security’s spending. “This weakens our national security, and our ability to maintain a strong and effective system to protect our country.”
Republicans were left shaking their heads at Democrats’ switch on Ms. Nielsen.
“It’s total hypocrisy,” Mr. Cornyn said.
Asked about Mr. Miller, Mr. Cornyn said he wasn’t sure what the advisor’s role was in the shakeup, but he said the White House does need a more consistent message on immigration.
“There are some voices in the White House who are working counter to that message and particularly creating more problems for the administration by losing senior leadership,” he said.
“A lot of the churn here is not helpful to [the president] or to DHS, but he’s the president, these people serve at his will. I don’t know what his rationale is, but it’s bound to create some turmoil,” Mr. Cornyn said.
Even before the shakeup, Mr. Trump had already been searching for someone to act as a border czar, coordinating policy across the government to deal with the record-breaking surge of illegal immigrants from Central America.
The most prominent name to rise is Kris Kobach, former Kansas secretary of state, who was under consideration for the Homeland Security job at the beginning of the Trump administration. Also mentioned as possible czar is former Virginia Attorney General Ken Cuccinelli.
⦁ Dave Boyer, Tom Howell Jr. and Bailey Vogt contributed to this article.