Sunderland boss Gus Poyet is still hoping to sign Marcos Alonso but admits a deal is 'close, but not close enough'.
Arsenal have announced the sale of Thomas Eisfeld to Fulham for an undisclosed fee.
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.