The Black Cats dismissed Paolo Di Canio after just six months in charge at the Stadium of Light following a start to the season which saw them fail to win in their opening five Premier League games.
Caretaker manager Kevin Ball will remain in charge until a permanent replacement is named and Short says there will be much consideration before any decision is made.
"Here at the club, like every supporter, we would like to see long-term success on the pitch and stability at the club," Short said in his matchday programme notes ahead of the clash with Manchester United.
"Unfortunately sometimes the quest for stability can be interrupted by the absolute necessity of staying in the league.
"This is because the long-term aim becomes irrelevant if we aren't at the top level.
"We now have a very important decision to make. Our only consideration in making that decision is what is best for the club. In doing so, we feel the weight of the entire city and beyond, along with the history of this massive football club.
"The appointment of a replacement head coach is our priority and I urge you to ignore the speculation in the media about the selection process, because most of it is completely wrong."