A 19-point deficit in the Premier League and a limp display at leaders Liverpool proved to be the final straw for United, who axed Mourinho after two and a half years on Tuesday.
Kovac has also found his position under threat this season with German champions Bayern nine points off the pace in the Bundesliga.
The Bundesliga side still sit inside the top four and have drawn Liverpool in the last 16 of the Champions League, with Kovac seemingly safe for the moment.
But the 47-year-old said he feels for Mourinho and hopes not to see any more managers sacked in the near future.
"I just want to say that I feel sorry about every manager sacking," Kovac told reporters.
"I can feel him, and my biggest wish is that no manager will be sacked anymore."
Manchester United has announced that Jose Mourinho has left the Club.— Manchester United (@ManUtd) December 18, 2018
We would like to thank him for his work during his time at Manchester United and wish him success in the future. #MUFC
Bayern are languishing in third position following three defeats from 15 league matches under Kovac, who is in his first season at Allianz Arena.
"We need a dominant and confident performance tomorrow. We know that we are on the way up. Tomorrow we need to bring the same strengths that we showed against Hannover." #FCBRBL #packmas #FCBayern #MiaSanMia pic.twitter.com/80P0WTX2XZ— FC Bayern English (@FCBayernEN) December 18, 2018
It was the first loss of the season for Dortmund, giving Kovac’s side a chance to close the gap at the top of the table to just six points with a win over RB Leipzig on Wednesday.
The club will then have just one more match ahead of the winter break, a Saturday showdown with Werder Bremen.