President Biden says that Minority Leader McConnell, who froze again at a press conference Wednesday, is “his old self.” Congresswoman Marjorie Taylor Green, though, is publicly saying he is unfit for office.

Mr. Biden said during a visit to the Federal Emergency Management Agency headquarters Thursday that symptoms like Mr. McConnell’s are not that uncommon after a severe concussion.

“I spoke to him today,” Mr. Biden said. “And you know, he was his old self on the telephone.”

The attending physician at the Senate, Brian Monahan, said in a statement Thursday that he had met with Mr. McConnell and his team and that the senator is “medically clear to continue with his schedule as planned.”

In a post on X, formerly known as Twitter, Ms. Greene said that Mr. McConnell was “not fit for office” after Mr. McConnell froze at a press conference in Kentucky on Wednesday, the second time he has done so this summer.

“Severe aging health issues and/or mental health incompetence in our nation’s leaders MUST be addressed,” Ms. Greene, so far the sole Republican suggesting that Mr. McConnell should retire, said. “Biden, McConnell, Feinstein, and Fetterman are examples of people who are not fit for office and it’s time to be serious about it.”

One Democratic congressman, Dean Phillips, joined Ms. Greene in publicly questioning Mr. McConnell’s fitness and called for term limits for members of Congress and the Supreme Court in the name of “basic human decency.”

“For goodness sake, the family, friends, and staff of Senators Feinstein and McConnell are doing them and our country a tremendous disservice,” Mr. Phillips said in a tweet.

On Wednesday evening, after the freeze-up, Mr. McConnell went on to attend a fundraiser with Congressman Jim Banks, who reported that Mr. McConnell was acting normally at the fundraiser.

“Enjoyed a wide-ranging discussion with Leader McConnell this evening about the future of our country,” Mr. Banks wrote in an X post. “I look forward to working with him on issues confronting our bordering states.”

Much of the conversation around Mr. McConnell’s health, though, has happened behind closed doors, with Politico and Punchbowl news both reporting that Mr. McConnell’s inner circle has met to discuss the issue, though even many of his allies aren’t in the loop about his health.

Mr. McConnell still appears poised to continue on through the 2024 elections, with Senator Scott, who has challenged him for his leadership position in the past, telling CBS that he expects Mr. McConnell to stay on.

“I expect he’ll continue to be the Republican leader through this term,” Mr. Scott told CBS. “We’ll have another election after the 2024 elections.”

While Mr. Scott doesn’t expect a leadership battle, it would only take five Republican senators to force a special conference meeting. According to Politico, some senators are discussing pushing for one when they return to Washington, D.C., in September.

Any move to remove Mr. McConnell would only stir a new set of challenges for the GOP Senate conference, as it would likely result in a power struggle between its different factions and could potentially end in the same way that Mr. Scott’s failed challenge to Mr. McConnell did after the 2022 election — a victory for Mr. McConnell.

The concerns around Mr. McConnell’s health also have some reviewing the laws around replacing a sitting U.S. senator in Kentucky, which were changed in 2021 by the state’s Republican-led legislature to give more power to the party and less to the governor.

Under the new law, Governor Beshear, a Democrat, would not be able to pick whoever he wants as a replacement. Rather, the Kentucky Republican Party would pick a slate of three candidates from their own party, and the governor would then select one of these three candidates.

After the appointment is made, a special election would be scheduled with an open, nonpartisan primary, and candidates would need to receive more than 50 percent of the vote to win. If no candidate meets that threshold, a runoff election would be scheduled.

While Mr. McConnell has shown no signs of planning to retire, his office did say that he would see a doctor about the issue. The relative silence about Mr. McConnell’s fitness from elected officials likely reflects the potential seriousness of the issue but also that questions of fitness for office are likely a two-edged blade for both parties.

While Mr. McConnell’s health has been making headlines in the past day, Senator Feinstein’s health and fitness for office was the subject of scrutiny for months earlier this year when she was absent from the Senate due to a bout of shingles.

Questions about Mr. Biden’s health, age, and fitness have been a perennial topic since before he took office and will undoubtedly be an issue through the 2024 election.