The executive chairman will make a decision on whether to remain at the club within the next month and manager Pochettino has confirmed if Cortese leaves, so will he.
Asked if he would be staying if Cortese left, the Argentinian said: "Of course not, because the person that has put his faith in me in this club and as a manager is Nicola Cortese.
"Nicola Cortese has basically been the reality at Southampton for the past four years.
"Of course, the people that are in the club, that are part of the club, that are behind the scenes in the club were a little bit worried about what has been said.
"Me on a personal sense, I would not understand a Southampton without Nicola being here."