McConnell stood by those remarks in a brief interview on Wednesday: “I don’t think I have anything to add to what I said about that earlier."

Cuccinelli, acting head of the relatively obscure U.S. Citizenship and Immigration Services, is one of the president’s top lieutenants because of his aggressive immigration agenda. But he is loathed by McConnell and other Republicans to the point that he probably could not be confirmed for a permanent job.

Sen. Roger Wicker (R-Miss.) said he has heard "doubts" about Senate Republicans' appetite to see Cuccinelli elevated and Sen. Shelley Moore Capito (R-W.Va.) said "he’d have difficulty being confirmed” to a permanent slot and that putting him in an acting role would be "a problem."

“We need somebody who knows the department really well. Somebody who understands the complexities and the realities of the challenges of balancing all that has," she said.

Trump has been pushed to give the job to Cuccinelli, a newcomer to the administration who has quickly developed direct access to the president, according to people familiar with the situation.

“If Ken was as liked by the White House as he thinks he is, he would’ve been nominated months ago," said a senior administration official. "The problem for Ken is that he is a nonstarter with Mitch McConnell and has a less than zero chance of being confirmed by the Senate.”

But Sen. Chuck Grassley of Iowa, the most senior Republican in the Senate, said under the Federal Vacancies Reform Act there's no legal way to put Cuccinelli in the position after acting DHS Secretary Kevin McAleenan leaves his post.

“There’s some opposition to Senate confirmation. I have not heard anything about some go-around. But it’s my understanding that the existing law would not permit him to” lead the organization, Grassley said in an interview. “I don’t know how you get around that. I don’t think it’s possible because of what the law says, not because of anything else.”

The administration is considering bypassing a federal law governing agency succession that makes Cuccinelli ineligible for acting secretary by naming someone to be assistant secretary of the Countering Weapons of Mass Destruction Office, which was recently vacated, and then elevating that person to be the acting secretary of Homeland Security, according to two people familiar with the situation. The loophole comes from language from a bipartisan bill that was inserted into the National Defense Authorization Act in fiscal year 2017, according to a person familiar with the situation. The New York Times first reported the effort to appoint Cuccinelli.

House Homeland Security Chairman Bennie Thompson (D-Miss.) on Wednesday called on the administration to disavow the maneuver.

“If the White House cannot find anyone qualified and suitable to run the Department of Homeland Security — or even run it in an acting capacity — something is very wrong with this administration,” Thompson said Wednesday morning.

The White House and Cuccinelli did not respond to questions Wednesday.

In response to a question, McAleenan told the House Homeland Security Committee on Wednesday that he doesn’t plan to rewrite the rules of succession, but declined to discuss any personnel issues. “The administration will follow the law in naming a successor for the Department of Homeland Security,” he said.

Some worry Cuccinelli is not the right fit for a department that tackles more than just immigration.

"Ken is liked by folks but unprepared to run DHS and its vast array of missions, from cyber security to counterterrorism," said a former senior administration official. "If he’s selected, senior officials at DHS and other security agencies have fretted that he will transform it into an all-immigration, all-the-time department, ignoring the other serious threats to the country he needs to pay attention to."

Asked about acting Undersecretary Chad Wolf, Grassley suggested he could be confirmed to the permanent position he has been nominated to and then take the acting job atop DHS. But it’s unclear that confirmation could happen quickly and Grassley said he had no inside information that Wolf will be Trump's pick.

Some lawmakers oppose Wolf because he was registered as a lobbyist for the National Association of Software and Service Companies, which represents Indian and U.S. companies that aim to keep the H-1B visa program for foreign workers with advanced degrees.

"People in the building [at DHS] are concerned about the legality and appearance of" doing an end run around the law to make Cuccinelli the acting secretary, said a senior administration official.

Not all Republicans are at odds with Cuccinelli. Senate Homeland Security Chairman Ron Johnson (R-Wis.) said he’s “enjoyed” working with him at USCIS and that he had no issue with his politics: “I don’t. I know some do. But that was in a prior life. You’ve got to look at how he’s performed in this particular role.”

As for whether it would be legal to appoint him to an acting role, Johnson said: “I’m not an attorney. I think there’s various opinions on that.”

Trump announced almost three weeks ago that McAleenan would step down as acting secretary after serving just six months. He is scheduled to leave Thursday but the administration official said he would be willing to stay an additional week.

The president has gone through four DHS secretaries in less than three years in part because he and his allies grew dissatisfied with officials they thought were not sufficiently committed to the president’s immigration agenda.

"God, I worry for the department," a former senior DHS official said. "Without consistent and responsible leadership, who is working to ensure that we aren’t vulnerable from a multitude of threats — cyber attacks, terrorism, natural disasters? Acting leaders can’t command the same respect with career employees or work with Congress effectively to protect the country."

Next in line to become acting secretary is Transportation Security Administration chief David Pekoske, who has also been serving as acting deputy secretary at DHS. But he has reportedly taken himself out of the running. Mark Morgan, acting commissioner of Customs and Border Protection, is also being considered, according to the people familiar with the situation.

Ian Kullgren contributed to this report.