The Daily Mail says Vardy plans to help the Rhinos, who have not competed since 2017 due to lack of funding, rejoin a league in 2022.
Vardy is believed to have been working on the project for two years as he sees similarities between Rochester – an industrial city about 300 miles north-west of New York – and his home town Sheffield.
"I've been looking at opportunities in the United States for a while, but there was something about Rochester and the Rhinos that just felt right," said Vardy.
"I love the story of the U.S. Open Cup win (when Rochester were the first club outside the MLS to do so 22 years ago), and although the club has been through some difficult years since, in a way that makes this more appealing to me.
"I can't thank Leicester enough for allowing me to do this. They know, and I know, that it will not in any way distract me from my priority, which is helping Leicester to be successful and scoring goals in the Premier League for many more years to come."