Rummenigge has already confirmed Milan's interest in 19-year-old Portuguese midfielder Sanches, who is facing a fight to break into Carlo Ancelotti's starting XI.
With Ancelotti able to call upon new arrivals Corentin Tolisso, James Rodriguez and Sebastian Rudy, Milan are reportedly ready to take Sanches on an initial two-year loan deal with the option to buy for €40 million in 2019.
And the Bundesliga champions are prepared to part with Sanches – who only made nine starts in 2016-17 – whether that be on loan or permanently.
"He won't get enough minutes at Bayern, so we're willing to negotiate a loan deal and maybe beyond that," Rummenigge said via Tuttomercatoweb.
"I'm not the coach, but I think Renato needs regular football. I can confirm Milan's interest, but talks didn't progress last week.
"If a club are interested in a Bayern player, they must be ready to deal with significant economic figures and I don't feel that Milan are ready to meet our demands.
"We'll say that it wouldn't be bad if a €48m offer came in for him."
Sanches came off the bench and played the entire second half in Bayern's penalty shoot-out loss to Arsenal in the International Champions Cup, after Wednesday's friendly ended 1-1.