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Flick revealed in a television interview after last week's win at Wolfsburg that he has asked the Bavarian giants to cancel his contract, which runs until 2023, at the end of the season.
Bayern's board subsequently voiced its disapproval at Flick for the timing of his comments, with a ninth successive Bundesliga title still not yet officially wrapped up.
However, amid talk of a possible stand-off between the two parties, Rummenigge hopes an agreement can be reached before Bayern's next game against Borussia Monchengladbach on May 8.
"We have agreed that we would sit down after the game in Mainz," he told Bild.
"If we are to meet Hansi's request, all parties have to work together to find a solution that FC Bayern is also happy with."
Flick has been strongly linked with the Germany job, which will become vacant when long-serving boss Joachim Low steps down after this year's delayed Euro 2020 tournament.
The 56-year-old last week declared that succeeding Low is "an option" he is considering, but no talks have been held with German Football Association (DFB) officials.
Asked if he has given any extra thought to his future plans ahead of Bayern's 2-1 loss to Mainz on Saturday, which delayed his side's title celebrations, Flick told Sky Sport: "I can't say.
"I don't know what the next week looks like. As coaches, we only have a rough plan for what we want to do for the next week, the next 14 days."
Meanwhile, Rummenigge reiterated that Bayern would never consider forming part of a breakaway division following the furore surrounding the European Super League fiasco.
Twelve clubs from England, Italy and Spain signed up to the league, but the plans are now in tatters - despite the best efforts of Florentino Perez and Andrea Agnelli - after the majority of those teams withdrew their support 48 hours later.
"Bayern are not up for such an event," Rummenigge said. "We all have a clear stance here. You cannot separate yourself from the football family by organising your own event at the expense of others.
"In the end, it might even have been a good thing that we saw this whole circus fall through within 48 hours, which put the whole world in an uproar.
"People have understood forever that football is something different than just business."