LOUISVILLE, Ky. — On the afternoon of the Kentucky Derby, Rand Paul, the state’s junior Republican senator and likely presidential candidate, spilled out of an elevator in the exclusive Jockey Club Suites of Churchill Downs with an entourage of women with flower-adorned hats, men in seersucker suits and Rupert Murdoch.

Mr. Paul’s guest was a special one. The libertarian brand of politics championed by Mr. Paul and his deep reservations about American intervention overseas have prompted more than a bit of wariness in The Wall Street Journal’s editorial pages, on Fox News and in other influential media outlets owned by Mr. Murdoch. For Mr. Paul, the would-be candidate, and Mr. Murdoch, arguably the most powerful broker in Republican politics, Saturday’s day at the races was filled with betting, losing, drinking and a long chat over kettle corn. It was part getting-to-know-you and part political audition, and marked a potential turn in the race for president.

That Mr. Murdoch, no novice when it comes to matters of political imagery, allowed himself to be paraded for six hours around the boisterous and bourbon-drenched grounds like a prize horse behind a proud jockey, amounted to a message to more establishment Republicans that, as Mr. Murdoch put it, “I’m very open minded.”

“I thought it would be fun to have him come down,” said Mr. Paul, who wore a powder blue herringbone blazer, pink tie and Ray-Bans for the occasion.