Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellPelosi: Trump hurrying to fill SCOTUS seat so he can repeal ObamaCare Senate GOP aims to confirm Trump court pick by Oct. 29: report Trump argues full Supreme Court needed to settle potential election disputes MORE (R-Ky.) is publicly signaling to President Trump Donald John TrumpOmar fires back at Trump over rally remarks: 'This is my country' Pelosi: Trump hurrying to fill SCOTUS seat so he can repeal ObamaCare Trump mocks Biden appearance, mask use ahead of first debate MORE not to pick former Virginia Attorney General Ken Cuccinelli to lead the Department of Homeland Security (DHS).

"I've not spoken to him about any of them. I have expressed my, shall I say, lack of enthusiasm for one of them … Ken Cuccinelli," McConnell told reporters during a pen-and-pad briefing in the Capitol on Thursday.

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Cuccinelli, the president of the Senate Conservatives Fund, has been floated as a successor to DHS Secretary Kirstjen Nielsen Kirstjen Michele NielsenDHS IG won't investigate after watchdog said Wolf, Cuccinelli appointments violated law Appeals court sides with Trump over drawdown of immigrant protections Democrats smell blood with new DHS whistleblower complaint MORE, whose exit Trump announced on Sunday. The conservative group has battled with McConnell and his allies in GOP Senate primaries, including backing Tea Party challenger Matt Bevin in McConnell's 2014 reelection race.

In addition to Cuccinelli, former Kansas Secretary of State Kris Kobach and acting DHS Secretary Kevin McAleenan are also reportedly in the mix to succeed Nielsen on a permanent basis.

Nineteen conservative groups sent Trump a letter urging him to pick the former Virginia gubernatorial nominee, saying they "have every confidence in his abilities; his tough on crime stance, his solution-oriented approach, his dedication to the rule of law, his love for America, and most importantly, his loyalty to the cause of making America great again," according to a letter obtained by The Washington Examiner.

Some Republican senators have publicly expressed opposition to Kobach, whose hard-line immigration stances could provide headaches to the administration if it were to nominate him.

Asked if Kobach could get confirmed, McConnell demurred.

"Look I'm not going to handicap all the people that could come up. There are a number of members … who have had some reservations about some of the names that have been mentioned," he said.

Republicans hold a 53-47 majority, meaning they could lose three senators and still confirm Trump's nominees, either with a tie-breaking vote from Vice President Pence or if they flipped a Democratic senator.