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The network rebuffed the incident, which Politico's Dylan Byers reported came in the summer of 2011 and involved Ailes asking Kristol to "take a negative stance toward a certain individual."
Byers reported, citing sources, that the incident "may have been a factor" - he also wrote that another factor that "may have played a role" was the network's courting of fellow conservative commentator George Will in October.
"Roger doesn't recall a specific or negative incident - he likes Bill and doesn't have an issue with him," a Fox News spokesperson told Business Insider in response.
"We made a lot of changes at the network and had been talking to George Will for some time. At this point, Bill's contract ran out and we declined to renew it."
Kristol didn't immediately respond to a request for comment, but he told Politico that he was always "treated nicely" at the network. He said he enjoys being a "free agent," and he has appeared on multiple networks, including Fox News, since his contract was not renewed as a contributor. He'll appear on the roundtable of NBC's "Meet the Press" Sunday.