Rangnick claimed they had the Paris-Saint Germain star "as good as signed" after the forward's father offered him to the then-second division club.
However, Rangnick said the deal hinged on him taking over as Leipzig coach, an assurance he could not give.
"We had as good as signed Kylian Mbappe (early in 2015)," Rangnick said in an interview with Bild, as reported by kicker.de.
"The problem was: at the time we had just parted company with (coach) Alexander Zorniger, that was just after the winter break in 2015.
"At that moment we didn't know who would become our coach and how it would continue.
"Kylian's father said to me, if I could say for certain that I would become the coach, then he would entrust the boy to me immediately. But I couldn't give him that promise in February."
In the end, Zorniger was replaced by Achim Beierlorzer until the end of the season, and when Rangnick did take over as coach, Leipzig won promotion to the Bundesliga in his only season in the role.
His replacement, Ralph Hasenhuttl, led the team to UEFA Champions League qualification last season.
Mbappe made his senior Monaco debut at the end of 2015 and has not looked back, becoming, at the age of just 18, one of the hottest properties in European football. He moved to PSG on loan in the summer.