But, a day after Bayern president Uli Hoeness labelled press treatment of the club a "joke", the headlines are kinder for Kovac thanks to Colombia star James, a player with whom he has reportedly shared a tense relationship.
On Friday, Bayern accused the media of "disrespect" and of breaking the German constitution with recent criticism in an extraordinary news conference.
The three key Bayern figureheads do not usually all face the media together, leading to reports the briefing is to address Kovac's future.
Kovac, 47, took charge in July, but heads into Saturday's game against Wolfsburg with the German champions without a win in their last four matches.
Bayern Munich have been stuck in a poor run of form recently and results have slid, with their most recent defeat coming in a 3-0 humbling at home to Borussia Monchengladbach.
Rummenigge even cited Article 1 of Germany's basic law: "The dignity of man is sacrosanct". "This applies to the media, it applies also to experts, and it applies above all to experts who once played football for this club".
Hoeness also slammed the media for its coverage. Former Bayern player Olaf Thon, now a TV pundit, said Boateng and Hummels were playing "old-man football" in the side's central defense.
"It's time that the most important club in Germany positions itself, " he said.
"We met on Monday after Germany's worldwide match against the Netherlands and have decided we'll no longer accept this style of journalism".
"I have no words for what I've read about Manuel Neuer, I would like to remind you that Manuel was world goalkeeper of the year four times", he intimated. When we played in Sevilla [in the Champions League quarterfinals last season], he was exclusively responsible that we almost were eliminated.