The 47-year-old's position has come under increasing scrutiny after his side's poor start to life back in the top flight after a seven-year absence.
Reports have suggested Cortese is set to replace Adkins and that the pair had showdown talks this week.
However, Adkins laughed off such speculation today 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.
"We've now had 10 games of the season and obviously only picked four points up. We sit at the bottom. That's fact, that's reality.
"As we always do at the football club, we assess situations. We have to accept where we are but then we look forward and move on.
"We're obviously looking at the next run of games. We take one game at a time in respect to the next one which is at the weekend against Swansea. We are focused on winning that."
There have been suggestions that failure to beat Michael Laudrup's side could lead to Adkins being sacked, but that is something he is 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.
"What we do is we get on. There doesn't need to be any talk about it. We will be a success here."
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.
It is a game that goalkeeper Artur Boruc could be involved in after the conclusion of an investigation into allegations he swore at home fans and threw a water bottle at them during the recent home defeat to Tottenham.
"We have had a thorough investigation into the situation against Tottenham," Adkins said.
"Obviously I didn't select him for the West Bromwich Albion game. We've had a thorough investigation that has now been concluded to the satisfaction of the football club.
"Artur is now back in contention for the remaining fixtures of the season."