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.
"Someone is always going to be the favorite to be sacked, and if you're bottom of the league, you should be the favorite probably," he told the BBC. "I'm not going to hide away from that. But what a great challenge we face."
Even though Adkins accepts speculation surrounding his future is inevitable, he insists he remains determined to steer his club clear of the relegation dogfight.
"I'm standing here today a far better manager than what I was when I joined Southampton," Adkins said to Sky Sports. "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."