Mr. Cain’s vulnerabilities have already prompted four Republican senators — Cory Gardner, Mitt Romney, Lisa Murkowski and Kevin Cramer — to announce that they would vote against him if he were nominated. Barring his winning Democratic support, this quartet would be enough to doom his chances.

Republicans are still pondering a challenge to Mr. Moore, who has raised alarms on professional as well as personal grounds (the I.R.S. claims he owes $75,000 in back taxes). He admits knowing next to nothing about monetary policy, and his opinion writing over the years has drawn criticism from economists of diverse views. As Justin Wolfers of the University of Michigan put it, “More than possibly any other economist in modern America, he has a track record of getting the big issues wrong.” Wall Street isn’t crazy about him, either.

Even more unsettling for Republicans has been the president’s spring cleaning at the Department of Homeland Security, spearheaded by his adviser Stephen Miller. Peeved that some of his policies have yet to be realized, like denying welfare benefits to legal immigrants and stripping child migrants of court-ordered protections, Mr. Miller has been targeting officials he considers insufficiently harsh. In recent days, the president has ousted a number of key officials, including Secretary Kirstjen Nielsen, whom he had long scorned as “weak” on border security.

To replace Ms. Nielsen, the White House has floated the names of such figures as Ken Cuccinelli, Virginia’s former attorney general, and Kris Kobach, Kansas’ former secretary of state. Both are immigration hard-liners. Mr. Kobach, who oversaw Mr. Trump’s failed voter fraud commission, has a reputation so toxic that multiple Republican senators have rushed to head off his nomination. John Cornyn of Texas, a member of leadership, declared that he could not support Mr. Kobach. Pat Roberts, from Mr. Kobach’s home state, proclaimed him unconfirmable — “Don’t go there,” he advised — and even Lindsey Graham of South Carolina, a chronic presidential pleaser, said this would be “a tough one.”

Mr. Cuccinelli has drawn the ire of none other than Mr. McConnell. The majority leader is said to have reminded his troops at a closed-door lunch last week that Mr. Cuccinelli heads the Senate Conservatives Fund, an anti-establishment political action committee that Mr. McConnell sees as having cost his team seats in past elections by championing unpalatable candidates. In 2014 , the group backed a Tea Party challenger to Mr. McConnell, a transgression not likely to be forgotten by the leader. Ever. Last Thursday, Mr. McConnell told reporters that he had expressed his “lack of enthusiasm” for Mr. Cuccinelli.