The Saints lost 2-0 to West Brom at the Hawthorns to remain rooted to the bottom of the table, having conceded an astonishing 28 goals in their first 10 matches.
But while Adkins accepts speculation surrounding his future is inevitable, he insists he remains determined to steer his club clear of the relegation dogfight.
"Someone is always going to be the favourite to be sacked and if you're bottom of the league, you should be the favourite probably," he told the BBC.
"I'm not going to hide away from that. But what a great challenge we face."
Speaking to Sky Sports, he added: "I'm standing here today a far better manager than what I was when I joined Southampton. I've got a hunger and desire to keep leading this football club.
"All I want is the best for Southampton. The supporters deserve that. I know there's been a lot of investment and there's a vision in place and we're all working hard to make that work. I'm proud and privileged to be the Southampton manager.
"At the moment, it's a little more challenging. I want to embrace it more than anything. I'm in the Premier League, I want to be here as long as my career allows as a manager and I want it to be with Southampton."
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Replays appeared to show that Peter Odemwingie handled the ball in the build-up to the opening goal, with the striker hinting at such in his post-match thoughts, but Adkins was keen to highlight his team’s lacklustre defending rather than a possible oversight from match officials.
Adkins continued: "[For the] the first goal, Mike Dean, the fourth official, is screaming down the microphone to the referee that it's handball four or five times but he's not picked it up, but the game's played on from there. We haven't had enough desire to close the shot down, it's got a deflection, we're dangling a leg, and it goes past Paulo [Gazzaniga].
"Our desire to defend our goal has to be better. Unfortunately this season we've conceded far too many goals and individually we can highlight things. We need to work hard on the training ground and get that stability with the back four and the goalkeeper so that we can get the platform to go on.
"We need a win, that's without question. To win you've got to go and score goals, and it's our endeavour to entertain and score goals. But you've got to keep them out at the other end. At the moment we're not doing a job keeping it out at one end."
West Brom manager Steve Clarke was delighted with his team’s resolve after conceding late winners in their previous two games.
The Baggies have risen above north London rivals Arsenal and Tottenham into fifth place, but the former Liverpool assistant has insisted that survival is all that is on his mind at the moment.
Clarke said: "We spoke before the game about making sure we got our season back on track with two good performances and no points. The performance wasn't as good as we've played but we controlled the game and got the win we wanted.
"We've said from the start that we focus on the primary objective which is to remain in the Premier League, that's always the first target. We're on the way to that and we have to build on that. The quicker we reach the safety margin then we'll look at other targets. "