The German side decided against signing the striker after concluding he was too old.
An impressive haul of 22 goals in 34 Premier League games this season has seen Vardy help bring the Foxes to within one win of their first ever league title, while also earning himself a spot in Roy Hodgson's England plans.
But the 29-year-old's career could have taken a completely different turn were it not for Leipzig's policy for signing new players, with Rangnick telling Sport Bild that the club chose not to sign the striker as they believed him to be too old.
"We desperately wanted to sign Joe Gomez, who later chose Liverpool, and we were flying back from London to Leipzig with his agent," he said.
"In the plane, the agent said to me 'Mr Rangnick, it's a crying shame that you are so radical about who you sign and only go for players who are under 24 because I've got somebody who would be just perfect for you. I can guarantee he would become a national team player under you, but he's already 27.'
"It was Jamie Vardy."
Leipzig, backed by Austrian billionaire Dietrich Mateschitz, currently lie second in the 2.Bundesliga table and are in a strong position for promotion to Germany's top flight.
But if they are to replicate Leicester's heroics and mount an unlikely title challenge, they will have to do so without Vardy, although it could have all been so different.