Reports have suggested Cortese is set to replace Adkins and that the pair had showdown talks this week. However, Adkins laughed off such speculation and insists his working relationship with the Italian remains strong.
"The relationship between me and the chairman is very good, which is an important one at any football club," Adkins said.
"I would imagine there would be loads of speculation about different things, but it is strong and it is good. We talk regularly as we have always done and we're determined to make this a success."
Asked if he has spoken to Cortese this week specifically about his position, he said: "There is no need to speak about my position.
"It is a football club determined to be successful. We obviously spoke in the summer about our strategies and we're working very hard."
There have been suggestions that failure to beat Michael Laudrup's side could lead to Adkins being sacked, but that is something he is not concerning himself with.
"Let's not hide away from the fact that in football at some stage there is always a parting of the ways between a manager and a football club," Adkins said.
"I am determined to be a success here at Southampton Football Club, the chairman is determined to be a success here at Southampton Football Club so we both share the same ambition."
Adkins was speaking at an unusually early pre-match press conference at the club's Staplewood training ground, which he revealed was because he is taking the players away for a few days before the Swansea match.
Source: PA
Source: PA