Pressure had been growing on Kovac all season despite his leading Bayern to the Bundesliga title and a DFB-Pokal triumph in his first year in charge.
Schweinsteiger played for Bayern between 2002 and 2015 before moving to United, where he played under Mourinho in the 2016-17 season.
He says Mourinho showed a keen interest in Bayern and the Bundesliga, as well as learning the German language in his downtime.
"I can imagine Mourinho in Germany," he told Bild. "I remember he was always asking me about Bayern and the Bundesliga. During our away games, there was always Bundesliga on TV.
"He really knew every single player, even from the smaller teams. He was also learning German.
"He has not worked in the Bundesliga, so I can well imagine that he would be tempted by a commitment in Germany."
Kovac's assistant Hans Flick will oversee Bayern's preparations for their Champions League clash with Olympiacos on Wednesday and Saturday's Der Klassiker against Borussia Dortmund.