Time is running out for Bayern and coach Louis van Gaal as the Bavarians cannot afford to drop any points in their Group A clash against Israel's Maccabi Haifa on Wednesday.
Bayern need to beat Haifa and then Juventus in Turin in December and hope the Italians do not overcome Girondins Bordeaux this week.
"We have to turn this corner quickly," Rummenigge told reporters. "If we don't succeed then we will have to look into it."
Dutchman Van Gaal, who took over in the close season in an effort to make the team more competitive in Europe, has so far failed to deliver and is under growing pressure.
Bayern, with only four points from four games, risk suffering an embarrassing and rare group stage exit and with influential midfielder Franck Ribery still injured and Arjen Robben doubtful, they have a mountain to climb.
"This is not about the coach but the club itself," defender Philipp Lahm said on Sunday. "If you're seventh in the (German) league then your self-confidence is at an ebb.
"Our game is hobbling along because we are insecure. Now we need a liberating victory."
Haifa, with no points or goals from four matches, still have an outside chance of overtaking Bayern with two wins in the final group matches to grab the Europa League spot.
Haifa, who lost 3-0 at home to Bayern in September, will be without suspended Brazilian midfielder Tiago Dutra, but are confident they can score their first goals of the campaign.
"Maybe the chances I missed are a sign we're saving our goals for the Champions League," said striker Shlomi Arbeitman, who has scored 10 goals in 10 domestic league matches but has failed to capitalise on a host of other opportunities.
"We are desperate to score our first goal and win our first points and it might just happen in Munich. Who knows?" he added.
Probable teams:
Bayern Munich: 22-Hans-Joerg Butt; 4-Edson Braafheid, 5-Daniel van Buyten, 6-Martin Demichelis, 21-Philipp Lahm; 17-Mark van Bommel, 31-Bastian Schweinsteiger, 44-Anatoliy Tymoshchuk, 23-Danijel Pranjic; 18-Miroslav Klose, 33-Mario Gomez.
Maccabi Haifa: 1-Nir Davidovitch; 27-Eyal Meshumar, 5-Jorge Teixeira, 21-Dekel Keinan, 17-Tsepo Masilela; 18-Ali Osman, 7-Gustavo Boccoli, 8-Jhon Jairo Culma, 20-Yaniv Katan; 9-Vladimer Dvalishvili, 19-Shlomi Arbeitman.
