Christopher C. Krebs, the director of the department’s cybersecurity agency, has also been floated as a temporary replacement for Mr. McAleenan but has no interest, administration officials said. Both Mr. Pekoske and Mr. Krebs have expressed to those close to them that they want to stick to their Senate-confirmed responsibilities rather than take up the acting secretary position.

Some White House aides are resisting Mr. Wolf as the choice because they believe he would be too lenient on immigration policy. Others in the White House are backing Mr. Wolf, who served as a top official for Kirstjen Nielsen in her tenure as homeland security secretary, and two administration officials described his record as strong.

While Mr. Cuccinelli and Mr. Morgan would face a difficult road to Senate confirmation, their fiery pronouncements and willingness to go along with Mr. Trump’s more extreme policies have pleased him. Last week, Mr. Cuccinelli received the public endorsement of the acting White House chief of staff, Mick Mulvaney, who said, “Ken would be good at the job.”

The Office of Legal Counsel, however, found that the next acting secretary must have served at least 90 days under Ms. Nielsen, the last Senate-confirmed person in the post. Mr. Trump forced her out, creating the vacancy of a Senate-confirmed chief. White House officials are also aware that elevating Mr. Cuccinelli could sow discord with Senator Mitch McConnell of Kentucky, the majority leader, who will have to manage Senate Republicans through a trial to determine the future of Mr. Trump’s presidency if the House impeaches him.

Mr. Cuccinelli endorsed an effort by hard-line conservatives to force Mr. McConnell to step down and backed Mr. McConnell’s primary challenger in 2014. Mr. McConnell previously voiced his disappointment to the White House over Mr. Cuccinelli’s appointment to the Department of Homeland Security. Mr. Cuccinelli told reporters last week that Mr. McConnell’s comments were “political.”

“I don’t know whether his view has changed, but certainly what I’ve been doing now is months removed from those comments you’re referencing,” Mr. Cuccinelli said, referring to his past as an anti-McConnell activist.