Flick on Saturday announced he wants to leave Bayern at the end of the season and has made no secret of his interest in succeeding Germany head coach Joachim Low.
Matthaus thinks Flick will land the Germany job, with Bayern moving for Leipzig's 33-year-old coach Nagelsmann.
Bayern and Germany legend Matthaus envisages Leipzig turning to Red Bull Salzburg's Marsch to fill the void if Nagelsmann is appointed as boss of the European champions.
He told Sky: "My opinion is: Flick to the DFB, Nagelsmann to Bayern and Jesse Marsch to Leipzig."
Matthaus added: "Of course, Bayern didn't speak to Julian Nagelsmann directly, but started the conversation with Julian Nagelsmann's side and I'm still standing by that today.