Vardy has scored 22 goals in 34 Premier League appearances for leaders Leicester this season, but the 29-year-old might have been playing in the German second tier had Rangnick refused to sign players over a certain age.
"We desperately wanted to sign Joe Gomez, who later chose Liverpool, and we were flying back from London to Leipzig with his agent," Rangnick told Sport Bild magazine.
"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 and he's also spent time in prison after a little fight.'
"It was Jamie Vardy."
Rangnick recalled himself replying: "No, he's 27 - we're not doing it."
Vardy instead stayed with the Foxes, scoring five goals in his first Premier League campaign before spearheading Leicester's unlikely title challenge the following season.
He has also scored two goals in six appearances for England.
Leipzig, who are backed by Austrian billionaire soft drinks tycoon Dietrich Mateschitz, are on course for promotion to the Bundesliga this summer.