The German giants parted company with manager Niko Kovac on Sunday following a 5-1 defeat by Eintracht Frankfurt with the club fourth in the Bundesliga.
Schweinsteiger, who worked under Mourinho at Old Trafford, has revealed that the former Real Madrid and Chelsea boss, who is the bookies favourite to take over in Munich, continually used to question him over Bayern, the Bundesliga and was even learning German during his time in Manchester.
"I can imagine Mourinho in Germany," Schweinsteiger told Deutsche Welle.
"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."
Mourinho this week was linked with Arsenal as Unai Emery faces an uncertain future at Emirates Stadium, though the Gunners have distanced themselves from the reports.