From his perch in the Ohio governor’s mansion, Republican John Kasich is keeping a keen eye on President Trump’s every move and showing a willingness — some say eagerness — to hurl rotten tomatoes toward the leader of his party when he gets upset with the act.

He has become a go-to anti-Trump source for the media and often doesn’t even wait for the press to call. Instead, he leaps in to pile on after presidential missteps.

While insisting he wants Mr. Trump to succeed, Mr. Kasich has pointedly refused to rule out mounting a primary challenge against the president in 2020. He said he wants to maintain a voice in the party because “you never know when duty calls.”

“I think what they are doing is exactly what I would do,” said Tom Rath, a New Hampshire based Republican Party strategist who advised Mr. Kasich in 2016. “He is preserving the option.

“This is not a party that typically throws out its incumbent — though we have never had a situation quite like this,” he said.

A poll released last summer found that Mr. Kasich held a 52 percent to 40 percent lead over Mr. Trump among likely Republican primary voters in New Hampshire — the same state that delivered Mr. Trump his first victory in the 2016 nomination contest.

The 2020 chatter picked up again this month after Jon Weaver, Mr. Kasich’s 2016 campaign manager, said the governor is not interested in challenging Sen. Sherrod Brown, a Democrat who is up for re-election in Ohio this year.

“To all the press calling, the answer is no. Bigger fish to fry. #TwoPaths,” Mr. Weaver said on Twitter, alluding to Mr. Kasich’s post-2016 memoir, “Two Paths: America Divided or United.”

Mr. Kasich’s office did not respond to repeated inquiries for comment, but plenty of people are awaiting his decision.

“Kasich is exactly right to say maybe it works out and maybe it doesn’t, but I am not going to spend every day of my life calculating the odds,” said Fergus Cullen, a former chairman of the New Hampshire Republican Party. He said he would immediately support Mr. Kasich.

Mr. Kasich refused to endorse Mr. Trump’s candidacy after being one of the last candidates standing against the New York tycoon in the 2016 Republican presidential primary — though that was more because of obstinacy than any surge of support. Mr. Kasich won just a single state: Ohio.

Since then, he has been a thorn in the side of the White House. In the past month, he has written op-eds criticizing Mr. Trump on immigration and foreign policy.

Last week, he piled on Mr. Trump over comments that the president reportedly said about Third World countries during an immigration negotiation at the White House. He demanded an apology from the president.

“These comments are inappropriate, and frankly the party ought to say it’s inappropriate,” Mr. Kasich told CNN.

Mr. Kasich called Mr. Trump’s travel ban ham-handed and said it offended Muslims. He also slammed the president’s response to racial clashes in Charlottesville, Virginia, last summer.

He urged young illegal immigrants to come to Ohio after Mr. Trump eliminated the Obama-era Deferred Action for Childhood Arrivals deportation amnesty program for illegal immigrant Dreamers, and he has warned that Mr. Trump’s “loose talk” about war with North Korea is dangerous and irresponsible.

On policy, Mr. Kasich has been one of the staunchest Republican supporters of Obamacare and even used his veto power to keep Ohio’s Obamacare-fueled expansion of Medicaid on track. He also argued against congressional Republicans’ efforts to repeal and replace the embattled health care law.

Mr. Kasich was lukewarm on the tax cut bill that Republicans have touted as their major accomplishment of 2017. He aired concerns about how the legislation would add to the national debt, which has been a pet issue of his since he played a chief role in balancing the federal budget as chairman of the House Budget Committee in the 1990s.

Ohio Republicans, meanwhile, said Mr. Kasich has grown delusional and is so far out of touch with the party’s base that he would struggle to win a primary race in his home state.

Should Mr. Kasich decide to take on Mr. Trump, he will be battling history: No candidate in modern history has defeated a sitting president for his party’s nomination.

There have been plenty of attempts. Former California Gov. Ronald Reagan came close in 1976 against President Ford, who assumed office after the resignation of President Nixon, and became the only president never to have been elected either president or vice president.

Craig Shirley, a Reagan biographer, said Ford was politically vulnerable after assuming the job because he inherited a bad economy and a mess in Vietnam from Nixon, and voters were not invested in him.

“Reagan had a perfect storm, and he still couldn’t beat Gerald Ford,” Mr. Shirley said.

Mr. Kasich, he said, is less known and less liked than Mr. Reagan, a former movie star who, prior to launching his insurgent bid, had twice-daily radio commentaries, twice-weekly columns and regular appearances on the TV talk show circuit.

“Reagan was a household name in 1975, and John Kasich — he could go on ‘To Tell the Truth’ and no one would guess him,” Mr. Shirley said, adding that people have made more of an investment in Mr. Trump than they ever had in Mr. Ford.

Sen. Edward M. Kennedy of Massachusetts attempted to defeat President Carter for the Democratic nomination in 1980 but also fell short. The most recent serious attempt came in 1992, when former top White House aide and journalist Patrick J. Buchanan challenged President George H.W. Bush in the Republican primary.

While none of three challenges against sitting presidents was successful, in all three cases the incumbent went on to lose the general election.

Mr. Cullen said Mr. Kasich’s decision on whether to run against Mr. Trump could hang in large part on the 2018 midterm elections.

“I don’t think the political situation or dynamic is fundamentally going to change until after the November elections,” Mr. Cullen said. “It is going to take Republicans getting defeated for a Republican to say, ‘We have to make a change.’”

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