Upamecano reportedly has a release clause worth €42.5 million ($51m), which Bayern has activated in principle.
"Dayot Upamecano will play for Bayern for the next five years," Salihamidzic told Bild. "We are very happy about it at FC Bayern."
"We had very good, intensive and professional discussions with Dayot and his advisor Volker Struth for many months. We knew that we had very strong competition," he added.
Bayern obtained this week their sixth title in 12 months after winning the Club World Cup,but are on the verge of losing defender David Alaba this summer after he rejected Bayern's offer of a new five-year deal.
"Upamecano is a young player, 22, whose qualities have already developed extraordinarily," Salihamidzic said. "I was always convinced that we had presented a good concept. We presented him with our vision of his career at Bayern. During the past week in Doha I had further talks with everyone involved. At the end of a long process, the players, family and management were convinced that FC Bayern was the right club."