Alonso recently announced he will hang up his boots at the end of the 2016-17 campaign after four seasons at the Allianz Arena, fuelling speculation they could enter the transfer market to bring in a replacement.
Rummenigge has hinted they may not splash the cash, though, after backing Sanches to fill the void left by the former Spain international.
"Our plan is for Renato Sanches to follow in Xabi Alonso's footsteps," the Bayern supremo told Sport Bild.
"That is the reason why we signed him ahead of this season.
"He definitely has the quality for it, although we might have to be a bit patient."