The Italian businessman was an integral part of Markus Liebherr's takeover in 2009 and stayed on after the Swiss billionaire died suddenly the following year.
However, having helped the club rise from the third tier to the Barclays Premier League, Cortese is understood to be considering his position due to frustrations over the backing he receives.
The executive chairman will make a decision on whether to remain at the club within the next month and record appearance holder Paine, appointed honorary president in December, has highlighted his importance.
"Southampton cannot do without Nicola Cortese," Paine told BBC Radio Solent.
"The chairman has made two or three brilliant decisions that may not have looked like it at the time but now it has come to fruition and we realise just what impact he has had and thank goodness for that.
"We do not want to be back to them old dark days where we were three years ago. He has taken us to where we are now by making some great decisions and some difficult decisions.
"We cannot rest on our laurels because he is the only one with the vision to take us further, that is why he has brought in the new manager. That is another part of his vision.
"He just can't leave. I believe if he leaves then in three years we will be back where we started [League One]."