They have collected only four points out of a possible 30 and conceded 28 goals - the worst defensive record in the top flight.
Adkins said: "I don't think anybody needs to give anyone a vote of confidence because if you do, then we all turn around and say, 'he's got the dreaded vote of confidence'.
"We need to win and I believe we can do it. I'm very proud and privileged to be the Southampton manager. I'll stand out and lead from the front as I've always done.
"I'm very positive. I know I'm a better manager today than when I joined Southampton. I know I've improved a lot.
"But someone is always going to be the favourite [to be sacked], and if you're bottom of the league, well, you should be the favourite probably.
"I'm not going to hide away from that. But what a great challenge we face."