Ohio’s secretary of state, Frank LaRose, is entering the race for the U.S. Senate, putting him in a competition with two self-funding primary candidates who are aiming to unseat Senator Brown, one of the most vulnerable Senate Democrats in 2024.

Mr. LaRose will face a state senator, Matt Dolan, and a businessman, Bernie Moreno, both of whom plan to contribute some of their own funds to their respective campaigns. No candidate has yet won the backing of the National Republican Senatorial Committee or President Trump.

“I’ve always believed with hard work and focus that anything is possible. But lately it feels like our country is slipping away from us,” Mr. LaRose said in an ad. “I’m a green Beret, a conservative, a man of faith, and I’m not afraid of a fight.”

Mr. Moreno was endorsed by Senator Vance, who defeated Mr. Dolan in the 2022 primary. Mr. Vance said in a statement accompanying the endorsement that he is hoping to avoid a “bloody primary.”

“It’s time to turn the tides on the establishment insiders who sell out our country to special interests and elect more political outsiders like Bernie, who will always put America First in Washington, D.C.,” Mr. Vance said.

Although Mr. Vance won his own primary last year with help from Mr. Trump, who endorsed him in April, Mr. Trump has yet to weigh in on the current primary. Mr. Moreno dropped out in February 2022, long before any primary votes were cast.

Mr. Moreno’s decision to drop out of the race in 2022 came shortly after he met with Mr. Trump and realized he wouldn’t be receiving the former president’s endorsement. After that, staffers disclosed to the Ohio Capital Journal that Mr. Monero didn’t see a path to victory and dropped out. 

Now, Mr. Moreno is drawing comparisons with Mr. Vance, promising to follow in his footsteps if he is elected to the Senate.

“The career politicians and establishment insiders in both parties have failed us all, and just like J.D., I will be a fighter for Ohio’s workers and families against the corrupt special interests in the swamp,” Mr. Moreno said.

Mr. Dolan has been more antagonistic to Mr. Trump and his acolytes, refusing to seek his endorsement in last year’s primary while every other major GOP candidate in the state sought the president’s favor.

Mr. Dolan’s refusal didn’t help him at the polls. He received 23.9 percent of the primary vote, coming in third behind Mr. Vance and the state treasurer, Josh Mandel, who also attempted to appeal to Mr. Trump’s followers.

Mr. LaRose’s relationship with Mr. Trump falls somewhere in between those of Mr. Moreno and Mr. Dolan. At a closed-door event in May, Mr. LaRose said he would expect to gain Mr. Trump’s support if he entered the race but that he wouldn’t be “begging for it,” according to reporting by Politico.

In Mr. LaRose’s assessment, only 20 percent of the party would vote for the candidate Mr. Trump endorses. That’s more than enough to swing a primary election, though. Last year, Mr. Vance won the nomination with only 32.2 percent of the vote in the primary.

“There is another 20 percent that care about who he endorses but that’s not going to be the decision maker,” Mr. LaRose said. “And then there’s probably another 60 percent of the party that doesn’t care who he endorses.”

Whoever emerges victorious from the GOP primary in Ohio will be facing off against Mr. Brown, a Democrat who national Republicans have long hoped to unseat in Ohio but who has held on since 2007.

A spokesman for the Ohio Democratic Party, Reeves Oyster, said that the primary would be damaging to the GOP field and that Ohioans “should ask themselves: What is Frank LaRose really doing for us?”

“The Republican ‘slugfest’ for Ohio’s Senate seat is shaping up to be another long, contentious battle that will leave whoever emerges damaged in the eyes of Ohio voters,” Mr. Oyster said in a statement.