Hodgson said Beckenbauer, a World Cup winner with West Germany both as a player and coach, felt the Englishman would have stood in the way of upcoming German coaches.
Hodgson's stock was high in international football at the time, having led Switzerland to the 1994 World Cup finals and to qualification for Euro 96.
Instead, Erich Ribbeck was appointed to replace Berti Vogts, who resigned following Germany's quarter-final elimination to Croatia at the 1998 World Cup.
Hodgson said that the German Football Association (DFB) president at the time, Egidius Braun, had been keen on appointing him.
"It was an honour when Braun called me," Hodgson told Germany's Bild am Sonntag newspaper.
"I was on the president's wishlist. He would have had problems convincing people to accept a foreign coach, though.
"At the time, my friend Franz Beckenbauer pointed out that a foreign coach of the Germany national team would not have helped the German coaching programme - the up-and-coming coaches."
After being brought out of retirement for an ill-fated two-year spell as Germany coach, Ribbeck admitted he was aware of the DFB's interest in Hodgson, and several other coaches for a position nobody seemed keen on taking.
Paul Breitner had been the preferred choice, but for a difference of opinion with Braun which led to Ribbeck becoming the oldest coach of Germany.
"If all of the names linked with the position after Berti Vogts and before me are true, then I was most likely only seventh choice," said Ribbeck at his presentation.
At the age of 61, he too did not fully match Beckenbauer's blueprint for a younger, German coach, a path which the DFB has since followed with Rudi Voller, Jurgen Klinsmann and, now, Joachim Low.
That search for youth and promise has extended to the players since Low took charge in 2006, and Hodgson admits the DFB has got it spot-on since Germany finished third as World Cup hosts seven years ago.
"The DFB's decision to focus on young players and install an aggressive style of play was exactly right," said the 66-year-old, who counts Germany among the favourites to win next year's World Cup, along with Spain, Argentina and the hosts.
England, meanwhile, are only "outsiders" for the next World Cup, but victory over Germany on Tuesday night is increasingly important after defeat to Chile on Friday night.
"The fans want to see a strong England team," Hodgson said. "It's all about prestige."