Cortese quit last week, leaving the future of manager Mauricio Pochettino in doubt, with the Argentine saying in May that he would follow the Italian banker through the exit door should he ever resign, although he has since calmed those fears.
"Funnily enough the last few days have been much the same as other weeks, even though there was all that talk of meltdown and such," Cork told reporters. "It has never felt like meltdown. We go in and train and play football like we always do.
"We were laughing at talk of us imploding. That was never going to happen. There's been a lot of silly talk.
"The gaffer came in and said: 'I'm not going anywhere. We carry on.' That was nice to hear. Mr Cortese was a good chairman but I don't think it's really going to affect any of us lads."
Southampton were forced to settle for a 2-2 draw at Sunderland on Saturday, despite leading by two goals at one stage of the match.