Kovac won his first seven matches in charge, in a blistering start to life since moving from Eintracht Frankfurt.
But a run of four games without a victory and just two goals against Augsburg, Hertha Berlin, Ajax and Borussia Monchengladbach has left him under pressure.
Speaking after the 3-0 defeat to Gladbach, the former Croatia captain said he was 'not the one to answer questions' on his future.
But a little over a year after Carlo Ancelotti was sacked after a run of 13 points from the first six league games of last season, club president Uli Hoeness backed Kovac.
"I will defend Niko Kovac to the blood," the president said, as quoted by ESPN. "It's all totally calm here."
However, with Arsene Wenger and Zinedine Zidane being linked to the Bavarian giants, former captain Lothar Matthäus deflected the issues off Kovac and laid into Rodriguez and the rest of the Bayern squad.
"James feels offended when he doesn’t start. Then he goes out and plays as if he were not part of the team," he told Sky Germany.
"The players have to come down from their pedestals and assume that they have to be at the service of the club, at the service of FC Bayern.
"Bayern is a world-class club and the players should be happy to play for such a great team."