Le Tissier, who was critical of Cortese's decision to sack manager Nigel Adkins and replace him with former Espanyol boss Mauricio Pochettino, had called for a meeting with Cortese.
The Saints legend later revealed talks had been scheduled for four weeks' time, but Southampton have said that those discussions are now off.
"A meeting was organised between Nicola Cortese, the club's executive chairman, and Matthew Le Tissier in what the club believed was good faith," a club spokesman said.
"Both parties had agreed that it would be a private meeting with no publicity surrounding it, yet within 24 hours this matter was put in the public domain and further negative comments attributed to Mr Le Tissier have appeared this morning."
The spokesman continued: "As an ex-player, Mr Le Tissier's achievements are well known.
"However, his repeated criticisms over a prolonged period of time are not justified and have not merited any response from the club. Nor has the motive for these personal attacks ever been made entirely clear.
"Mr Cortese wishes to reiterate that he will not make comment about somebody he does not know personally - and that includes Mr Le Tissier.
"These criticisms have not affected the club's success and progress on or off the pitch in recent years and will not divert the club from its course in the future.
"For Mr Le Tissier to agree to meet 'in good faith' and then make such comments can only be seen as a breach of good faith and certainly not in the best interests of the club.
"Mr Le Tissier also refers to his friend Francis Benali having the best interests of the club at heart, while neglecting to mention that the same person is currently suing Southampton Football Club in relation to a lease dispute and lost ambassadorial fees. It is a legal action the club will be defending rigorously in the best interests of the club and its reputation."
Le Tissier told Press Association Sport that the meeting's cancellation "doesn't surprise" him and wrote on Twitter: "Meeting with our wonderful exec chairman been cancelled #surprisesurprise #somethingtohidenicola." Benali was unavailable for comment.