On Sunday night, it was claimed that Milan had opened talks with Bayern over a potential loan deal for Sanches, who failed to score in 25 appearances for his German club during the 2016-17 campaign.
Rummenigge has revealed that there is genuine interest from Milan in the 19-year-old, but "no agreement" has been reached.
"I can confirm that AC Milan is interested in Renato Sanches. No agreement, however, has been reached," Rummenigge told reporters.
It is understood that Milan want to sign Sanches on loan until the end of next season, with the option to purchase the midfielder for £35m on a permanent deal in next summer's transfer window.
Sanches starred for Portugal during their Euro 2016 triumph last summer, before joining Bayern in a big-money move from Benfica, rejecting the chance to join Manchester United in the process.