On Monday morning, as Vivek Ramaswamy was soft-launching his campaign for governor of Ohio, Dan Merenoff sat at a coffee-shop counter in Delaware, a suburb of Columbus, weighing the prospect of Mr. Ramaswamy, a political celebrity but a governing novice, running the state.

“I like his views,” said Mr. Merenoff, 44, an operations manager for a pallet company. But he also liked Mr. Trump, and was not sure what to make of Mr. Ramaswamy’s exit from Mr. Trump’s Department of Government Efficiency task force two weeks earlier.

“I’d still like to read a little bit more, because he was backed by Trump for a little bit, and then all of a sudden that changed gears,” he said.

Mr. Ramaswamy, 39, is expected to formally announce his campaign for governor late this month, according to a spokesman, a move that follows a brief appointment to the second Trump administration that ended before Inauguration Day. He has never occupied a government office in Ohio, or even run for one. He moved his investment firm out of the state, to Texas, last year.

Still, as long as Mr. Ramaswamy is a Trump ally in good standing, he appears to have a leg up.

To many Ohioans, he needs no introduction after two years of dogged scrambling across the landscape of Republican politics, a blur of Fox News hits and county committee meet-and-greets that has made him one of the party’s most visible figures.

“There’s a lot of momentum behind him,” said Barbara Orange, the executive chairwoman of the Lucas County Republican Party.

In an interview on Friday, Dave Yost, the state’s attorney general and Mr. Ramaswamy’s main rival for the nomination — whose forthcoming book has a promotional blurb from Mr. Ramaswamy on the cover — pointed to his depth of government experience in the state, and to Mr. Ramaswamy’s lack of it.

“I’ve got results,” he said, “and he quit on the first day of his one federal job.”

But political observers in the state generally agree that Mr. Ramaswamy’s fate most likely turns on whether he receives the endorsement of Mr. Trump, whose backing has become a decisive factor in statewide races across the country.

Mr. Ramaswamy, who declined to comment through the spokesman, built his political brand on the critique of environmental, social and governance investing that he advanced in his 2021 book, “Woke, Inc.” After dropping his presidential bid last year, he became an indefatigable champion of Mr. Trump on the campaign trail.

He is running for governor only after finding himself at odds with Mr. Trump’s administration, but he is also presenting himself as an extension of many of its goals, telling The Wall Street Journal that if he is elected, he will focus on trying to “shred the regulatory barriers to new business creation” in Ohio and put the state on the “bleeding edge of a new kind of industrial revolution.”

Greg Moss, a retired farmer and construction worker in Kingston, Ohio, said that as a fan of Mr. Ramaswamy, he looked forward to voting for him for governor. But Mr. Moss added that Mr. Trump’s endorsement was important to him. He would vote for “everybody he endorses, hands down — it doesn’t matter.”

The singular weight of Mr. Trump’s endorsement is a familiar story in most states by now, but it is still a relatively new one in Ohio, where Republican politics remained stubbornly old-school well into the Trump era. The party’s nominations for prominent offices have tended to go to well-established politicians who have woodshedded for years in county and lesser state offices, and built relationships with local power brokers across the state’s 88 counties.

The state’s term-limited governor, Mike DeWine, is a relatively moderate figure by current Republican standards. In 2020, he broke from Mr. Trump — who endorsed him in 2018 only after he had won the primary — over Covid-19 restrictions. Mr. DeWine nonetheless beat back a primary challenge from the right in 2022.

But this old order was battered in the state’s 2022 and 2024 Senate races, when Mr. Trump’s endorsement propelled JD Vance and Bernie Moreno, both political newcomers, to primary victories over better-established rivals from the state party’s moderate and right wings alike.

“We now have these MAGA outsiders who’ve done nothing in state politics getting in front of people who’ve been in state politics for years,” said David Pepper, the former chairman of the state’s Democratic Party.

Mr. Ramaswamy would seem to fit a similar bill. Like Mr. Vance, whom he has known since they were classmates at Yale Law School, he grew up in the state but has little record in Ohio politics. By his own account, he did not vote in presidential elections between 2008 and 2016, and was registered to vote as an independent as recently as 2021.

The governor’s mansion is the latest of several offices sought by Mr. Ramaswamy during his brief time in politics. He publicly mused about running for Senate in 2022, and after he dropped his presidential bid in early 2024, speculation abounded that he would be in line for a cabinet post in a future Trump administration.

But instead, he was named along with Elon Musk to help lead the so-called Department of Government Efficiency, the para-governmental task force that has upended Washington in recent weeks with its rapid infiltration and drastic transformation of the federal bureaucracy.

Mr. Ramaswamy reportedly clashed with Mr. Musk and some Trump aides, and in the days before Mr. Trump took office, both he and the president began exploring possible exit paths.

Mr. Trump encouraged Mr. Ramaswamy to seek appointment to Mr. Vance’s Senate seat, and Mr. Ramaswamy met with Mr. DeWine to discuss the matter in mid-January. But Mr. DeWine chose his lieutenant governor, Jon Husted, for the seat. Three days after that announcement, sources close to Mr. Ramaswamy said he was leaving Mr. Trump’s task force to run for governor.

There are indications that the decision was a quick one. The domain name vivekforgovernor.com was registered by an unknown party only days after Mr. Ramaswamy’s meeting with Mr. DeWine. When Mr. Ramaswamy joined a live conversation with Mr. Musk on X this month, Mr. Musk seemed to be struggling to keep up with his political ambitions, wishing him well as “perhaps one day Senator Ramaswamy — or Governor Ramaswamy.”

“Yeah, that’s kind of the direction we’re headed,” Mr. Ramaswamy replied.

A poll circulated in late January by Tony Fabrizio, the Trump-aligned pollster who also worked for Mr. Vance’s Senate campaign in the state, found Mr. Ramaswamy with a 34-point lead over Mr. Yost for the nomination.

Unaligned political observers in the state considered that finding credible, and in interviews across several Ohio counties last week, Republican-leaning voters were broadly supportive of his candidacy.

“He’s not a career politician, and 99 percent of what we have now is corrupt,” said Pam Minardo, 62, who works filling orders for Instacart and other delivery services in Delaware County, where Mr. Yost built his political career. “Columbus needs to be cleaned up.”

Bryan C. Williams, the chairman of the Summit County Republican Party and a former acting chairman of the state party, said Mr. Ramaswamy’s position most likely reflected his outsize profile in national politics.

“The first poll is not the last poll,” he said, noting that the primary election was over a year away.

Still, Mr. Williams said he had recently advised other candidates in the race that “if Vivek Ramaswamy declares, he’s an immediate front-runner.”

The field has already thinned in advance of his announcement. Robert Sprague, the state treasurer, had filed papers to run for governor, but last week he announced he was running for secretary of state instead and endorsed Mr. Ramaswamy.

The only Democrat to join the race so far is Amy Acton, the former director of the Ohio Department of Health.

Supporters of Mr. Yost argue that the attorney general, who sued the Biden administration over a range of federal policies in recent years, has a path to Mr. Ramaswamy’s right, and suggest that Mr. Ramaswamy’s quick departure from his Trump appointment makes his endorsement by Mr. Trump a less-than-done deal.

Mr. Yost has hired Justin Clark, a strategist and lawyer who served on Mr. Trump’s 2020 campaign team, as an adviser.

“I am very hopeful to earn his endorsement,” Mr. Yost said of Mr. Trump. But he added, “Anybody that tells you that they know what the president is going to do before he announces it is probably spinning you.”

Still, Mr. Ramaswamy’s nascent campaign team includes several strategists close to the White House, including Mr. Fabrizio, Jai Chabria, a political strategist who worked on Mr. Vance’s Senate campaign, and Andy Surabian, a longtime adviser to Mr. Trump and Donald Trump Jr. — a strong indication of Mr. Trump’s approval of his bid.

Ryan Stubenrauch, a Republican strategist and a former senior policy adviser to Mr. DeWine, suggested that a matchup between Mr. Ramaswamy and Mr. Yost, a three-decade veteran of state and local government, stood to reveal what, if anything, remained of the old ways of the state’s Republican politics in the age of Mr. Trump.

“If it’s a two-person race, it’ll be the ultimate test,” he said. “Thirty years in politics, the years of experience that used to be required — is that over and done with?”