"I'm gutted. We came here to win. The defeat and especially the way we played hurts," said Rummenigge after the third straight loss for the defending Bundesliga champions consigned them to fourth place, 19 points off the pace.
"We have had an absolutely catastrophic eight days prior to this... but the way we played today, you have to say, that was the absolute low point of our season," he added.
Asked about van Gaal's future, he said: "Of course we are all disappointed, but I think we have all been around long enough to know that after such a game, you need to have a night's sleep before you consider what to do.
"You need a bit of rest to consider what you can do, what you should do, to stop this rot. We are in a situation now that is of great concern for Bayern Munich."
"Actually, I won't sleep but I will think."
Earlier, van Gaal refused to speculate over his future, saying only: "I am not in a position to answer this question.
"I have always had the support of my board. They decide whether to withdraw this support, not me."
Rummenigge said the powers-that-be would not sit down immediately with van Gaal to discuss the recent run of form but that they would take at least a day to mull things over.
Van Gaal said he could not fault his players' efforts in the second period against Hanover, but ultimately they came up short.
"It's very difficult to take but in the second half, our players played their hearts out, but it just wasn't enough.
"We started well and their first shot resulted in a goal. After that goal, the pressure was really on us."
The result also left Bayern five points shy of second place and automatic Champions League qualification.
Before the Hanover game, the German giants had endured a torrid few weeks in which their dreams of repeating last season's league and cup double were cruelly smashed.
First they suffered the humiliation of losing 3-1 at home to runaway title leaders Borussia Dortmund, then four days later, they were dumped out of the German Cup thanks to a 1-0 semi-final defeat to Schalke.
Hugely off the pace in the league, their hopes of salvaging the season lie in securing second place as well as in the Champions League itself where they hold a 1-0 lead over holders Inter Milan, who beat them in the final last season.
Some fans have already started to turn against their coach, with signs demanding 'Van Gaal Out!' seen at Munich's Allianz Arena after Wednesday's cup defeat.
But he appeared still to have the support of the dressing room, with captain Philipp Lahm saying: "The team is sticking by the coach."
The Dutchman himself, who formerly coached Ajax, Barcelona, the Netherlands and AZ Alkmaar had said that "Everything hangs on this result" before the Hanover match.
He is under contract until June 2012 and has no plans to quit Bayern having taken over in 2009 after the sacking of former Germany coach Jurgen Klinsmann.
He has admitted he has been approached by other clubs without naming them.
Finally, asked if this was the most difficult period in his professional career, he said: "No, that was the second period at Barcelona."