Senate Minority Leader Mitch McConnell‘s on-camera health scare set off a political guessing game in Kentucky about how the Democratic governor will name a replacement if he gets the chance.

The guessing game in Washington was about who is on deck to lead Senate Republicans if Mr. McConnell goes.

Kentucky Governor Andy Beshear finds himself boxed in by the 2021 law requiring the governor to file mid-term vacancies in the U.S. Senate from a list of three candidates chosen by the state party of the previous sitting senator.

Mr. Beshear may thumb his nose at the new law and try to name a Democrat.

“I would imagine you would absolutely see a lawsuit on this,” Michael Abate, a Louisville attorney who’s worked for the Kentucky Democratic Party in the past, told the Lexington (Kentucky) Herald-Leader.

“Beshear either says, ‘Hey, Republican Party, thank you for your list, but I’m appointing whoever I want’ and then that immediately gets challenged in court, or you could see Beshear taking the route of filing a lawsuit,” Mr. Abate said. “I mean, he’s got the guts to defy it.”

In Washington, the question was who will succeed Mr. McConnell as the Republican leader, a post he has held for 17 years. Likely contenders include:

  • Sen. Rick Scott of Florida who last year unsuccessfully challenged Mr. McConnell for the job;
  •  Senate Republican Whip John Thune of South Dakota;
  •  Republican Conference Chairman John Barrasso of Wyoming;
  •  Sen. John Cornyn of Texas, a former member of the Senate GOP leadership team who retains leadership stature in the conference.

The jockeying for the job goes on quietly behind the scenes in the Senate. Any public moves to line up votes in the GOP conference will wait until Mr. McConnell signals his departure, which he isn’t doing yet.

In Kentucky, Mr. Beshear set the stage to challenge the 2021 law when he declared it unconstitutional and vetoed it. The legislature’s Republican supermajority overrode the veto.

In a veto message, the governor cited the U.S. Constitution’s 17th Amendment passed in 1912, which states that the legislature “may empower the executive” to make temporary appointments to fill a senate vacancy.

“The bill … upends a century of precedent by delegating the power to select the representative of all Kentuckians to an unelected, unaccountable committee of an organization that represents only a fraction of Kentuckians,” Mr. Beshear wrote. “The Seventeenth Amendment does not authorize legislatures to direct how the Governor makes an appointment to fill vacancies, and the legislature may not impose an additional qualification on who the Governor may appoint beyond the qualifications for a United States Senator set forth in the Constitution.”

Mr. McConnell has insisted all these parlor games are for naught, saying, “I’m fine.”

Despite the frightening episode when Mr. McConnell, 81, appeared dazed and unable to speak for about 20 seconds Wednesday at a Capitol Hill press conference, his office said he plans to finish his term that ends in 2026.

In a statement, his office said Mr. McConnell appreciates the continued support of his colleagues and “plans to serve his full term in the job they overwhelmingly elected him to do.”

Mr. McConnell might not have to hold on until 2026 to safeguard the replacement process for his Senate seat. Kentucky will elect a governor in November when Mr. Beshear faces Republican Daniel Cameron, who now serves as the state attorney general.

Kentucky Senate President Robert Stivers cautioned not to make political plans based on speculation about Mr. McConnell‘s health.

“I take Leader McConnell‘s staff at face value. If they tell me he is fine, I have no reason to doubt them. I fully expect Leader McConnell to continue in his role, and all of the speculation suggesting otherwise is an absurd attempt to create a story out of nothing,” he said in a statement to The Washington Times.

Republican Party activists say the Kentucky GOP has long prepared for Mr. McConnell‘s departure from the Senate and have told The Times his deteriorating health has long been a concern for those in his inner circle.

“It is widely rumored at the [state] capitol that Senator McConnell was in poorer health than what was commonly known until [Wednesday],” said Lee Watts, a radio talk host and pastor who holds prayer meetings at the state capitol. “It was expected that he would likely resign if we are to get a Republican governor this November.”

Meanwhile, Mr. McConnell‘s relationship has soured with some of the state Republican Party, with several county parties voting on censure resolutions against him and others calling on him to resign over various issues.

Some party officials see enough evidence now to support Mr. McConnell stepping down.

“Age and ability start to weigh in there somewhere,” Jerry Bird, the GOP chairman in LaRue County, Kentucky, told The Times. “You have to say to yourself, ‘Am I capable of doing this?’”

Potential replacements floated for the list of three include state Rep. Josh Calloway and Rep. James Comer, who currently chairs the House Oversight and Accountability Committee.

Mr. McConnell, who earlier this year suffered a concussion and a fractured rib after a fall, froze up mid-sentence at the Wednesday press conference and stared unresponsively for about 20 seconds before other Republican senators stepped in to ask if he was all right.

He did not answer, but slowly walked back to his office with an aide and Sen. John Barrasso of Wyoming, a former orthopedic surgeon who is the No. 3 Republican in the Senate.

Mr. McConnell returned after about 10 minutes and then fielded questions on a variety of issues.

“I’m fine,” Mr. McConnell said.

Asked if he is still able to do his job, he said, “Yeah.”

Even as McConnell tried to brush off the concerns, the episode raised new questions among his colleagues about his health and also whether McConnell, who was first elected to the Senate in 1984 and has served as Republican leader since 2006, might step aside from his leadership post in not from his Senate seat.

McConnell‘s aides previously denied any lingering health problems from his recent fall, which resulted in a six-week hiatus from the Senate.

Mr. McConnell and his top deputies chuckled when asked if there was a successor in mind to replace him as the Republican leader eventually.

At the start of the new Congress in January, Mr. McConnell broke the record for the longest-serving Senate leader, which was previously held by the late Democratic Sen. Mike Mansfield of Montana. Mansfield served as majority leader for 16 years.