Kristol, the editor of The Weekly Standard magazine and a leader of the conservative movement to find an alternative to likely major-party nominees Donald Trump and Hillary Clinton, confirmed to CNN Special Correspondent Jamie Gangel on Tuesday that the search has zeroed in on French.
French is a writer for another conservative publication, National Review, and a constitutional lawyer. He's also an Iraq war veteran and a Bronze Star recipient.
His writings have been sharply critical of Trump.
"The party of Lincoln is in ruins. A minority of its primary voters have torched its founders' legacy by voting for a man who combines old-school Democratic ideology, a bizarre form of hyper-violent isolationism, fringe conspiracy theories and serial lies with an enthusiastic flock of online racists to create perhaps the most toxic electoral coalition since George Wallace," French wrote in March
French on Wednesday dismissed doubts that his candidacy could succeed.
"All the normal political rules apply. The conventional wisdom has been right. An underdog can't win. Right?" French tweeted.
Kristol has increasingly engaged in a war of words with Trump
over whether the search, which has also involved 2012 GOP nominee Mitt Romney and conservative blogger Erick Erickson, would ultimately produce a candidate.
"I know David French to be an honorable, intelligent and patriotic person. I look forward to following what he has to say," Romney tweeted on Tuesday.
French himself was until a week ago pushing Romney to run.
"You're the only man who can save us from future calamity," French wrote in a May 24 National Review post
, "Mitt Romney, Run For President." French argued a Trump-Clinton general election matchup would be "a choice between two historically corrupt, dishonest, and incompetent politicians."
Romney, French wrote, "is the only man who combines the integrity, financial resources, name recognition, and broad public support to make a realistic independent run at the presidency."
Before news of a potential French bid broke on Tuesday afternoon Trump, speaking at a news conference in Trump Tower in New York, Trump repeatedly called Kristol a "loser."
He said The Weekly Standard publication was "failing" and questioned Kristol's political judgment.
"He's been wrong for two years," Trump said.
Kristol soon fired back on Twitter.
"I gather Donald Trump said I'm a loser. I've won some and I've lost some, but one thing I've always tried not to be is a roaring jackass," Kristol tweeted Tuesday afternoon.