Kessie is in the final year of his deal with the Rossoneri and has been linked with the likes of Chelsea, Arsenal and Liverpool.
The 24-year-old, currently representing Ivory Coast at the Games in Tokyo, has assured Milan he has no intention of moving on.
"I'm proud to have chosen Milan and it is not my intention to leave. Indeed, I want to stay forever," he told Gazzetta dello Sport:
"When I come back from the Olympics, I'll fix everything. I just want Milan. Sporting director Paolo Maldini knows my thoughts.
"I want to be the 'president' of Milan for life."
Milan head coach Stefano Pioli said he has no concerns about Kessie departing earlier this month.