The developments at St Mary's put the future of Southampton boss Mauricio Pochettino in doubt, with the Argentinian having previously said he would leave Saints if Cortese left.
Sky sources indicate that Cortese handed in his resignation some time ago following a power struggle with Liebherr.
Cortese, a former banker, fronted a deal on behalf of Markus Leibherr to buy Southampton in 2009 when the club was in administration and in League One.
Since then, the value of the club has dramatically increased, following back-to-back promotions to the Premier League.
Katharina is the daughter of Markus, who died a in 2010. She now wants to sell the club, whilst Cortese wants to wait.
This process is believed to have been ongoing since last summer - it was understood that a compromise agreement was reached only for relations to break down again over Christmas.
Pochettino's future is now unclear. Back in May, Pochettino stressed that his involvement at Southampton was dependent on his chairman remaining at the club.
"If (he) Cortese doesn't work here...I would not understand a Southampton with Nicola being here," he said.
"The person who has put his faith in me at Southampton is Nicola Cortese."
There has so far been no comment from Southampton