Spain World Cup winner Alonso spent three seasons at Allianz Arena between 2014 and 2017 before the cultured midfielder called time on his distinguished playing career.
Alonso won three Bundesliga titles, a DFB-Pokal and the DFL-Supercup during his time at Bayern and is now in charge of the B team at Real Sociedad, the club with whom he rose through the youth ranks and started in the professional game.
Rummenigge remains in contact with Alonso and foresees a time when the Spaniard is back at Bayern, this time in the dugout.
"He will be a coach who may be of interest to FC Bayern at some point in the future. I can imagine that," Rummenigge told the BILD podcast 'Phrasenmaher'.
"We are still friends today, he reports regularly.
"Xabi Alonso has this empathy that you need. Especially with today's generation of players to spin such a common thread with them."
Rummenigge hopes such a scenario does not occur too soon, though, given the success enjoyed under Hansi Flick.
Former Germany assistant Flick led Bayern to a Bundesliga, Pokal and Champions League treble last term and Rummenigge is confident of a long association between club and coach.
"I hope he stays [a] long, long, long [time]. This is a coach who for the first time I trust will stay long," he added.
"Three to five years, that would be an extraordinary period."
Among the other topics discussed were who Rummenigge felt were the most important transfers of his tenure in the boardroom.
Unsurprisingly, Germany goalkeeper Manuel Neuer and superstar striker Robert Lewandowski were highlighted as such transfers.
"Manuel Neuer and Robert Lewandowski were the most important transfers," he said.