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Bayern said the 22-year-old would sign a five-year contract but did not reveal the size of the transfer fee. German media reported that Upamecano had a €43 million buyout in his Leipzig contract.
"We're happy that we've been able to sign Dayot Upamecano for FC Bayern Munich," Bayern sports director Hasan Salihamidzic said. "We're convinced that Dayot will be a very important member of our team in the coming years."
"We are happy to have been able to convince Dayot Upamecano to come to FC Bayern," Salihamidzic said. "Dayot will be a very important element for our team in the years to come."
Upamecano, who has three France caps, had also been linked with Premier League clubs Liverpool and Chelsea.
"We knew that we had very strong competition," Salihamidzic said. "Upamecano is a young player, 22, whose qualities are already exceptionally developed."
Bayern won the Club World Cup on Thursday to claim a sixth trophy in less than a year but are likely to have defensive vacancies by the summer when both David Alaba and Jerome Boateng are out of contract.
Alaba has made clear he plans to leave and, at 32, Boateng does not expect to be offered another contract.
Upamecano joined RB Salzburg at the age of 16 and moved to Leipzig 18 months later. He has played almost 140 matches for the club and helped them reach the Champions League semi-finals for the first time last season.
"We presented him with our vision of his career at FC Bayern," Salihamidzic said, adding that negotiations had continued while he was in Doha for the Club World Cup. "At the end of a long process, players, family and management were convinced that FC Bayern is the right partner."
Bayern already has a strong French connection with internationals Kingsley Coman, Corentin Tolisso, Lucas Hernandez and Benjamin Pavard, as well as the former Metz and Marseille player Bouna Sarr and 18-year-old Tanguy Nianzou, a product of the Paris Sain-Germain youth system who joined last summer.