(CNN) Republicans in Ohio and Washington, DC, are scrambling to find a strong candidate for Ohio's key Senate race, with two men -- a Republican representative and a top-selling author -- eyeing the seat in the battleground state.

J.D. Vance, who gained national recognition for his 2016 book, "Hillbilly Elegy," is seriously considering a run, an adviser to Vance told CNN on Wednesday. This comes after Senate Majority Leader Mitch McConnell spoke with Vance about his run, according to sources with knowledge of the call, offering the author advice on what he could expect from a potential bid.

At the same time, the White House's top political adviser, Bill Stepien, met with Rep. Jim Renacci on Wednesday at the White House. Renacci, who has been running for governor in Ohio, has floated the idea of switching to a Senate run and sources tell CNN that he expects to officially announce his Senate bid on Thursday.

Renacci told an Ohio radio station on Monday that he would drop the gubernatorial bid to run for Senate if President Donald Trump asked him to.

The decision last week by Ohio Treasurer Josh Mandel to bow out of the Senate race has Republicans looking for a strong candidate to challenge Sen. Sherrod Brown, the incumbent Democrat.

Vance, who graduated from Yale Law School and enlisted in the Marine Corps, has considered stepping into the vacuum.

"J.D. is giving serious consideration toward this, because there are very serious people asking him to run," Jai Chabria, a Vance adviser, told CNN.

Chabria added that "people are starting to realize (Vance) has the best message to beat Sherrod Brown," adding that "the phone hasn't stopped ringing since Friday" with people asking him to get in.

A source close to Vance said the author was in Washington on Wednesday and met with Republicans eager for him to challenge Brown. The source added that Vance is aware that he has a short window to make a decision -- the filing deadline is the first week of February -- and that the author is receiving a considerable push from Republicans in Ohio.

Vance was opposed to Trump's 2016 campaign -- he voted for an independent candidate -- and is not close to the President or his political orbit. The best-selling author does have ties to Republican Sen. Rob Portman of Ohio, the Koch Brothers and others within the conservative community, a source familiar with Vance's electoral plans told CNN.

Renacci, though, would be the candidate closer to Trump's orbit, and he has said he would need the White House's backing before getting in the race.

"My goal is to be the governor of the state of Ohio," Renacci told WTAM radio in Cleveland on Monday. "If the President of the United States reaches out and contacts me and asks to me to jump in that race, I would consider it only at that point."

Republicans, though, already have a crowded field in the Ohio governor's race and Trump's top political aides are interested in ensuring a strong candidate challenges Brown. Mike DeWine, Ohio's attorney general, and Mary Taylor, the lieutenant governor of Ohio, have already declared bids to succeed Republican Gov. John Kasich.

The Ohio Senate race is less crowded. Mike Gibbons, a businessman and Republican donor, has declared his bid. His campaign responded to the possibility of Renacci running by trying to stay as close to Trump as possible.

"No one has done more to support President Trump in Ohio than Mike Gibbons has, including serving as a campaign finance chair for the President and no one will do more than Mike to advance conservative policies and the Trump agenda in the Senate," Gibbons' campaign general consultant Mike Biundo said in a statement.

The White House official said Wednesday's meeting with Renacci "should not be viewed as an endorsement or White House support" and that the congressman did not meet with Trump.

Brown told reporters Wednesday in Ohio that he was not going to weigh in on his primary opponent.

"I run on my record," he said, adding that he doesn't think "the voters of Ohio should have Mitch McConnell decide who the Republican candidate for the state of Ohio will be."

Vance has not responded to CNN's repeated requests for comment.

Since publishing his memoir of growing up in the Rust Belt, which received considerable attention after Trump's win in 2016, Vance has flirted with the idea of running for office.

He told The Atlantic in 2017 that he considered running for Senate but decided against it, saying, "It would have been an objectively bad call for my family."

NBC's Megyn Kelly also asked Vance during a 2017 interview whether he would run for office in the future.

"I'm very flattered when people ask me," he said. "You never say never. But it is just not something that I think about doing right now."

CORRECTION: An earlier version of this story misstated the year "Hillbilly Elegy" was released.