The Munich chairman says the Bundesliga giants made an approach to bring the Frenchman to Bavaria in the mid 1990s.
The now Arsenal boss instead decided to join Japanese side Nagoya Grampus, where he he was at the helm between 1995 and 1996.
When Bayern's approach for Wenger was rebuffed, they instead appointed current Hertha Berlin boss Otto Rehhagel, who masterminded Greece's successful 2004 European Championships campaign.
Speaking to the Daily Mail, Rummenigge said: "In England, there are many good coaches and I like Arsene.
"We were very close in the 1990s to signing him. He was managing Monaco. We were in discussions to bring him to Munich, but he decided to go to Tokyo."
Rummenigge also spoke of how impressed he was with the work of Pep Guardiola so far at Bayern.
He added: "'We said this was a unique chance to get the best and most popular coach in the world, and we could not miss this chance.
"He is obsessed about football, always thinking, 'what can I do new?' to surprise the opponents - or to surprise his own team.
"We have had good coaches in the past, very successful ones. He is a different story in the positive sense."