With John Boehner now departing as House speaker, an influential Republican Party official is now seeking the ouster of another GOP leader who has frustrated conservatives: Senate Majority Leader Mitch McConnell.

“McConnell needs to resign!!” Louisiana GOP Chairman Roger Villere wrote in a Facebook posting.

Mr. Villere isn’t just any Republican. He’s the longest-serving state GOP chairman in the nation, with 12 years on the job, and is the vice chairman of the Republican National Committee, the GOP’s national governing body. He also serves on the RNC’s executive committee that makes decisions alongside Chairman Reince Priebus.

“Mitch is a good and honorable guy, but the base is leaving our party,” Mr. Villere said in an interview with The Washington Times. “I’m out in the field all the time and we have all our elections this year for state offices, and it’s hurting us tremendously with our elections.”

Aides for Mr. McConnell, Kentucky Republican, did not return repeated calls and emails seeking comment. A spokeswoman for Mr. Priebus said he was unavailable Sunday.

Egged on by irate rank-and-file GOP voters, a conservative rump group in the House GOP caucus forced Mr. Boehner’s sudden announcement on Friday that he will resign his post and his House seat as well next month.

An ideological irony is that his probable replacement, California Rep. Kevin McCarthy, the current House majority leader, is closer politically to Mr. Boehner than to the more ideologically passionate members who were prepared to call for a no-confidence vote on Mr. Boehner as speaker.

The same may hold true for Mr. McConnell in the Senate. Though he is not considered likely to resign, there is no one in his chamber whom conservatives seem ready to unite behind as an alternative.

Even Mr. Villere was stumped when asked whom he preferred to take over as leader of the Senate Republican majority.

“Honestly, I haven really thought of a replacement,” he said. “We are being so beat up by the base. I just was frustrated.”

Mr. Villere did say what specifically about Mr. McConnell makes his state’s rank-and-file GOP voters so dyspeptic that they want him out as leader — his failure to challenge executive overreach by President Obama or fight to repeal Obamacare and other unpopular measures.

“Mr. McConnell could have suspended consideration of confirmations for all presidential appointees, except for those who are essential to national security, until the president rescinded his unconstitutional executive action on amnesty,” Mr. Villere said.

“This would have been a constitutionally appropriate response to the overreach of the executive branch,” he said. “It would have transformed the political environment, greatly encouraged Republican donors and grass-roots activists, and positioned us to refuse to confirm replacements for any Supreme Court openings that might occur during the remainder of the Obama administration.”

Mr. Villere ticked off other sources of discontent with Mr. McConnell.

“Not trying to repeal Obamacare, not defunding Planned Parenthood, not trying to stop illegal immigration,” he said. “That is what Republicans ran on and once they were elected they did not follow up with their promises.”

Louisiana Gov. Bobby Jindal, speaking at the annual Values Voters Summit in D.C. on Saturday, also called on Mr. McConnell to resign, in what political observers chalked up as an attempt by Mr. Jindal to gain national publicity for his sagging 2016 presidential campaign. Elections for his successor as governor will take place this November.

Mr. Villere’s term as chairman will expire this coming March. He said he wasn’t aware that Mr. Jindal had said at the Values Voters Summit that it was time for Mr. McConnell to go.

Mr. Villere, while hearing the complaints of GOP base voters, said he was moved to act after reading a story quoting Mr. McConnell as saying he wasn’t going to support a potential government shutdown in order to block federal funding for Planned Parenthood, a private organization that provides abortion services.

“If we lose the battle, we will never win the presidency again in my lifetime,” said Mr. Villere, who is 66. “I’ve worked for 12 years as chairman to build this party, and I just don’t want to see it all go down the drain because they aren’t willing to fight for what we believe in. Our base is demanding we do something or they’re going to leave us.”

“The GOP brand is being damaged. We’re having to work from scratch to do all the groundwork we’ve already done with elections of Republicans to Congress and in the state because everybody is so furious at the leadership,” Mr. Villere said.

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