Bayern Munich sacked Kovac on Sunday after the German champions were humiliated 5-1 at Eintracht Frankfurt a day earlier.
Already a host of big names have been linked with the job at the Allianz Arena including the unattached trio of Jose Mourinho, Massimiliano Allegri and Arsene Wenger as well as Ajax manager Erik Ten Hag and Ralf Rangnick.
Matthaus, who won the first ever FIFA World Player of the year in 1991, has talked up the chances of the leading candidates for the job but has reserved special admonition in the case of Mourinho.
“I believe that Bayern will seek and appoint a coach who has a bigger name and can show more titles,” Matthaus told Sky Deutschland.
"The names are Jose Mourinho, Massimilano Allegri, Erik ten Hag, Ralf Rangnick and Arsene Wenger. To be honest, I could imagine them all at FC Bayern. They are great names, great coaches and everyone brings something very interesting.
“The question that only a few can answer, however, is: What does Bayern want? Where do the German champions want to go? What do you want to stand for? What would the bosses on the field want to see in the future?
“Mourinho I would rule out. I admire him and everything he has achieved and won in the past with many clubs. But I believe that his character and his way of working does not fit Bayern and a lot of unrest would prevail. But it would be spectacular, especially for the media.
“Wenger would have some advantages. He could easily communicate with the large faction of French players (similar to Arsenal) at Bayern and with all the Germans and the bosses. He's 70 years old, but at a similar age to Jupp Heynckes who also enjoyed great success. Certainly that would not be a solution for many years.
“Ralf Rangnick recently proved in Leipzig that he can create a perfect balance between defensive and offensive. His football is attractive and successful. Whether in the past things have stuck between Rangnick and Bayern, or the RB past would be a problem for fans, I cannot judge.
“Massimiliano Allegri? If Bayern does not necessarily want a coach who speaks German, he would be the most exciting solution for me. A great name, incredibly successful with Juventus, a gentleman and football man of the very highest category and he could easily talk to Bayern boss Karl-Heinz Rummenigge in Italian.
“Erik ten Hag? If it is important to the bosses that the new guy speaks our language, then I would make the Ajax coach first choice. His team plays wonderful offensive football. He has thrown Juve and Real deservedly out of the competition in the last Champions League and only by a miracle did not reach the final. In addition, he has already won some national titles.
“Ten Hag knows Bayern, because he has already trained there, is 49 years old and could be a man for the present and future. He has helped players like [Frenkie] de Jong and [Matthijs] de Ligt to become world class.”
