Jonathan Ernst/Reuters
Who could it be? Mitt Romney, the 2012 GOP nominee? Nebraska Sen. Ben Sasse, a fierce Trump critic?
But the potential candidate Kristol, the editor of The Weekly Standard, was revealed to be courting did not measure up as a big political name.
The supposed "impressive" candidate with "a real chance" is reportedly David French, a conservative lawyer, veteran of the Iraq war, and National Review columnist, as Bloomberg's Mark Halperin and John Heilemann reported Tuesday.
Romney himself, however, seemed to indicate a willingness to be supportive of French.
"I know David French to be an honorable, intelligent and patriotic person," he tweeted Tuesday night. "I look forward to following what he has to say."
French is relatively unknown outside of entrenched conservative circles. His wife, Nancy French, who has ghostwritten for 2008 Republican vice presidential nominee Sarah Palin and US Olympian Shawn Johnson, said her husband is still on the fence about hopping into the race.
"He's been trying to get other people to run for a long time, so that's probably why people are considering him, you know, because he's been very adamantly anti-Trump, or at least Never Trump, Never Hillary," she told NBC News on Wednesday.
She added: "So we just want someone to run. I don't know if it'll be him - it'll be quite jolting to our lives if that is the case. But he'd be great. We just don't know if it's the right thing for him to do."
David French/Twitter
One person that Halperin and Heilemann reported was "deeply involved" in the efforts to recruit an independent candidate said the search was centered on candidates who had one of the following three traits: fame, "vast" wealth, and experience in elected office.
French has not held elected office and is not of great fame or known to hold vast wealth.
"All the normal political rules apply," he tweeted Wednesday. "The conventional wisdom has been right. An underdog can't win. Right?"
Presumptive Republican nominee Donald Trump has ripped into Kristol since the weekend, tweeting that he's a "dummy" and later calling him a "loser" at a Tuesday press conference.
Senate Majority Leader Mitch McConnell also weighed in, telling Business Insider in a Tuesday interview that he hoped Kristol failed in his attempt "because that will help elect Hillary Clinton to the presidency."
"I like Bill Kristol and I know he's a smart guy, but I hope he's unsuccessful," the Senate majority leader said.