Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
QPR loanee Tom Carroll wants to "make it at Tottenham" but is prepared to consider all of his options at the end of the season.
The Italian was appointed to the post in 2009 after being instrumental in the purchase of the south coast outfit that year by Swiss businessman Markus Liebherr, and Cortese has since overseen the Saints' rise from League One to the Barclays Premier League.
It is understood that Cortese is contemplating quitting due to divisions between him and Katharina Liebherr, who became the club's owner after the death of her father Markus in 2010.
However, reports that Cortese submitted his resignation in writing are understood to be incorrect.
Southampton have not made any official comment on the matter.