Both magazine Kicker and daily Bild report that Bayern will pay €43 million ($68 million) to activate the buy-out clause in Upamecano's Leipzig contract, which runs until 2023.
European champion Bayern appears to have fought off competition from Liverpool and Chelsea, which also wanted to sign the centre-back.
Upamecano will replace David Alaba, 28, who is set to leave Bayern when his contract expires in June after repeatedly turning down extension offers.
Bayern sports director Hasan Salihamidzic told Bild he could confirm the transfer of the 22-year-old Upamecano.
"I can and we at FC Bayern are very happy about it," Salihamidzic said.
"We had very good, intensive and professional talks with Dayot and his agent Volker Struth over many months.
"We knew that we had very strong competition.
"We presented him with our vision of his career at FC Bayern."
Salihamidzic said the transfer details were finalised while Bayern was in Qatar, where it beat Tigres in Friday's (AEDT) FIFA Club World Cup final to claim a sixth title in the past year.
"During the past week in Doha, I spoke once again with everyone involved," Salihamidzic added.
"At the end of a long process, players, family and management were convinced that FC Bayern is the right partner."
Upamecano started on the bench in Saturday's (AEDT) Bundesliga 2-1 win at home to Augsburg, but Leipzig sports director Markus Kroesche stopped short of confirming the transfer.
"Basically, it's a fact that he has a buy-out clause and we aren't in the driver's seat, I can't say more than that," Kroesche said.
He insisted that Upamecano was rested because of Wednesday's (AEDT) UEFA Champions League Round of 16, first leg tie against Liverpool in Budapest.