Former Germany star Klinsmann was relieved of his duty only after 10 months of his appointment, as he failed to keep the German champions on an even keel.
Now former Barcelona coach Rijkaard, who was also sacked after a poor run with the Catalans last season, has become a candidate for the Bayern hot seat.
"One solution would be Rijkaard, he doesn't have a contract at the moment," Beckenbauer said on German broadcaster Premiere.
"It is still a little early to talk about this, but, of course, one must think about possibilities.
"We have to look into who would fit at Bayern and then make our decision."
Other Candidates
Luiz Felipe Scolari, Roberto Mancini, Bernd Schuster and Martin Jol are all believed to be considered by the Bayern board alongside Dutch tactician Rijkaard.
Chelsea interim boss Guus Hiddink and Arsenal manager Arsene Wenger were once linked with the vacant seat, but Beckenbauer believed none of the pair will take up the job.
"I value him (Hiddink) over all others, but he will leave Chelsea to concentrate on the Russian team," the former Germay star commented. "He won't take his place at Bayern."
On the possibility of hiring Wenger, Beckenbauer said: "Naturally he is an option, but we will not get him.
"He has developed the side in England and will not want to leave."