The Saints have endured a tough start to the season and are rooted to the bottom of the Premier League after winning just one of their opening 10 games of the campaign.
Adkins is confident he can turn the situation around though and has insisted he has the full support of the board at St Mary’s Stadium.
He told reporters: "The relationship between me and the chairman is very good, which is an important one at any football club. I would imagine there'll be loads of speculation about different things but it's strong and it's good.
"We talk regularly as we've always done and we're determined to make this a success."
When asked if he had spoken to Cortese about his position he said: "There's no need to speak about my position."
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He added: "In football, at some stage, there is always a parting of the ways between a manager and a football club. I'm determined to be a success here at Southampton Football Club, the chairman's determined to be a success here at Southampton Football Club, so we both share the same ambition.
"And what we do is we get on. There doesn't need to be any talk about it. We will be a success here.
"Any team who can stay in the Barclays Premier League after just being promoted - that's a great achievement, and it's one that we are determined to achieve."