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The defender, who will leave RB Leipzig to join Bayern Munich at the end of this season, was targeted by United as a teenage prospect when he was with French club Valenciennes.
He elected to turn down United and move to Austrian club Salzburg, Upamecano says, after being wowed by Rangnick's precise vision for his career.
At the time, Rangnick was sporting director for Salzburg and Leipzig, with both clubs part of the Red Bull empire, and talking Upamecano into a move proved one of many successes he achieved in that role.
United had tempted the youngster, however, and Upamecano told The Athletic: "I was a young boy. I wanted to sign for them straight away. Manchester United! But then my parents took me aside and said: 'Let's think carefully about this'.
"We thought about it for a long time and then decided that going 'etape par etape' [step by step] was the best option for me. Everything Ralf said came true."
Upamecano was loaned out to Liefering before returning to make an impact with Salzburg while still in his teens, and at the age of 18, in January 2017, he was moved through the ranks to join Leipzig.
Rangnick, who had already enjoyed one spell as head coach of Leipzig, returned for another in 2018, and Upamecano said: "Ralf was the most important coach for me. He's a professor, a scientist of football, and he also cares about everything. He's always there for you with advice on matters football and non-football, every single day."
Before joining Bayern, Upamecano has the chance on Thursday to land a DFB-Pokal title with Leipzig as they head into the final in Berlin against Borussia Dortmund.
Julian Nagelsmann, who succeeded Rangnick in 2019, has been another major influence on the career of Upamecano, who describes the 33-year-old as "an unbelievable manager".
Like Upamecano, Nagelsmann will also move to Bayern at the season's end.
"He really takes you to another level," Upamecano said. "For example, because of Julian, I have improved a lot in terms of opening the game from the defence with precise passes into the midfield, and he told me to use my dribbling abilities in order to create spaces going forward."
That is borne out by data that shows Upamecano has progressed significantly while at Leipzig.
Following his mid-season arrival in 2017, Upamecano had an overall passing accuracy of 78.6 per cent in the Bundesliga, while his accuracy with passes ending in the final third of the field was just 46.2 per cent.
These numbers belong firmly in Upamecano's past, because his precision now shows a marked improvement.
In 2017-2018, he stepped up those percentages to 83.7 and 55 in the Bundesliga, while attempting 130 passes into the final third of the pitch.
But in 2019-2020, his overall accuracy stood at 89.1 per cent and his accuracy into the final third was a healthy 74 per cent, with Upamecano attempting 232 balls into that attacking area in league games. Each figure put him in the top 10 for Bundesliga defenders who played at least 20 games.
This season has seen a minor step back, but an 88.5 per cent total accuracy and 70 per cent accuracy into the final third continues to reflect well on the French defender, a player United must sorely regret failing to recruit.