WASHINGTON — Senator John Thune of South Dakota, the second-ranking Senate Republican and a potential future leader of the party, is seriously considering retiring after next year, a prospect that has set off an intensifying private campaign from other Republicans urging him to seek re-election.

Mr. Thune is only 60, but a combination of family concerns and former President Donald J. Trump’s enduring grip on the Republican Party have prompted the senator, who is in his third term, to tell associates and reporters in his home state that 2022 could be his last year in Congress.

His departure would be a blow to South Dakota, which has enjoyed outsize influence in Washington, and could upend Senate Republicans’ line of succession. Mr. Thune has been open about his ambition to lead his party’s caucus after Senator Mitch McConnell makes way, and quiet but unmistakable jockeying is already underway between Mr. Thune and Senators John Cornyn of Texas and John Barrasso of Wyoming.

“John is the logical successor should Mitch decide to not run again for leader,” Senator Susan Collins of Maine said of Mr. Thune, while noting that Mr. McConnell’s hold on their caucus remained “very secure.”

That Mr. Thune would even entertain retirement with the chance to ascend to Senate Republican leader illustrates both the challenges of today’s Congress and the shadow Mr. Trump casts over the party.

Part of Mr. Thune’s hesitation owes to Mr. Trump and the potential for the former president — who lashed out at Mr. Thune early this year when the senator rejected his attempts to overturn the election — to intervene in South Dakota’s Senate primary race. But the larger factor may be the longer-range prospect of taking over the Senate Republican caucus with Mr. Trump still in the wings or as the party’s standard-bearer in 2024.

Mr. Thune has said he will decide his intentions over the holidays. Yet a number of his friends and colleagues have become convinced that he is serious about leaving public life.

Among those alarmed is Mr. McConnell himself, who one adviser said had “leaned in” on pushing Mr. Thune to run again.

“I certainly hope that he will run for re-election, and that’s certainly what I and others have been encouraging him to do all year long,” Mr. McConnell said in an interview.

He is hardly alone.

A range of Senate Republicans — from moderates and Trump targets like Ms. Collins to Trump allies like Kevin Cramer of North Dakota — have lobbied Mr. Thune over dinner, on the Senate floor and, since lawmakers went home for Christmas recess, via text messages.

“I let him know how much I appreciate him,” said Mr. Cramer, who has known Mr. Thune since they were young executive directors of their state parties. “He knows both Dakotas really need him.”

Mr. Thune first angered Mr. Trump during the former president’s final days in office, when Mr. Thune said any challenge to the election results “would go down like a shot dog.” Mr. Trump derided the senator as “Mitch’s boy” and urged Gov. Kristi Noem of South Dakota to run against him in the state’s primary.

Since then, though, Mr. Trump has trained his fire on Mr. McConnell, whom he has labeled “Old Crow,” and largely ignored Mr. Thune.

Two top Senate Republican allies of Mr. Trump said he would probably refrain from targeting Mr. Thune simply because the senator, who is popular at home and has a well-stocked campaign war chest, is unlikely to lose a primary in the state that first elected him to Congress in 1996.

“He likes winners, and John Thune is a winner,” said Mr. Cramer, predicting that Mr. Trump would at most be “a nuisance” to Mr. Thune.

Senator Lindsey Graham of South Carolina was similarly blunt about why Mr. Thune need not sweat a competitive primary. “Trump worries about his win-loss record,” said Mr. Graham, who is the de facto liaison between the former president and Senate Republicans.

Mr. Graham, who along with Mr. Thune and Ms. Collins is part of a small group of senators who often dine together in Washington, said that before they left for the holidays, he had reassured Mr. Thune about any Trumpian intervention.

“I told John that’ll be fine,” Mr. Graham recalled. “John will be fine.”

Asked if he thought the threat of a Trump-inspired primary bothered Mr. Thune, Mr. McConnell said, “No. No, I don’t.”

But if Mr. Thune ascended to Republican Senate leadership, Mr. Trump could still prove a headache.

The former president does not have the influence in the Senate, where 19 Republicans defied him to support the infrastructure bill, that he does in the House. Yet Mr. Trump’s regular attacks on Mr. McConnell and on anything that has the air of cooperation with President Biden are not lost on Senate Republicans.

A handful of them whose seats are up in 2022, including Mr. Thune, opposed the infrastructure bill after the former president’s relentless criticism of the bipartisan measure made it difficult for Senate leaders to back the legislation.

Perhaps more significant regarding Mr. Trump’s future influence is the turnover in the Senate and the question of whether retiring mainstream Republicans, like Senators Richard Shelby of Alabama, Rob Portman of Ohio and Roy Blunt of Missouri, will be replaced by Trump acolytes.

“We’ve just got to plow through this to the post-Donald Trump era, which I believe is coming,” Ms. Collins said, lamenting that the former president’s “haranguing the leader, Mitch, has gotten worse lately.”

If Mr. Thune left, she said, she would “truly be beside myself.”

Echoing Ms. Collins, if not as unequivocally, about why Mr. Thune should stay, Mr. Graham and Mr. Cramer both said he could eventually succeed Mr. McConnell, who will be 80 in February. Mr. Cramer said that Mr. Thune’s ascension would not happen “by default” but that “it would be really good for the farm belt.”

Mr. Thune has privately expressed confidence that he will have the votes to become leader whenever there is a vacancy, according to Republicans who have spoken to him.

In an interview before the recess, Mr. Thune told Punchbowl News that the possibility of becoming leader was “obviously a factor in considering whether to take another run,” adding, “It’s something I’m interested in.”

Hoping to limit the window for any Trump interference and competition in the primary, which is scheduled for June, Mr. Thune has put off making a final decision.

Many Senate Republicans thought that was but a formality.

He has, however, become increasingly candid about the temptation of returning to his home in Sioux Falls, where he has young grandchildren.

In an impromptu interview this month that unnerved his supporters in South Dakota, Mr. Thune said running again would mean at least six more years of commuting to Washington — a lifestyle for which his wife, Kimberley, has little enthusiasm. “She is done with it,” Mr. Thune told a local journalist.

If Mr. Thune retires, it would represent a striking historical symmetry in South Dakota. The state’s two other best-known senators, George McGovern and Tom Daschle, both Democrats, also served three terms and left the chamber at a moment when they enjoyed enormous clout.

But both lost their re-election bids. Mr. Thune would be leaving voluntarily.

Carl Hulse contributed reporting.