But Adkins insists he is in a positive mood for the challenges ahead and is a better manager than when he first took charge at St Mary's.
The Saints' latest setback was Monday's 2-0 defeat by West Brom at the Hawthorns - via two goals from Peter Odemwingie.
They have collected only four points from a possible 30 and have the Premier League's worst defensive record with 28 goals conceded.
When told he was the bookmakers' favourite to be sacked, Adkins said: "I've been in football all my life so you know what happens.
"I know I'm a better manager today than when I joined Southampton and when I was at Scunthorpe.
"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 and I'm full of positivity for that challenge ahead."
Adkins added: "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."
Albion are up to fifth place in the table after bouncing back from injury-time defeats against Manchester City and Newcastle.
But Baggies boss Steve Clarke is refusing to get carried away with a points tally of 17 from the first 10 games.
Clarke said: "Seventeen points from 10 games is good and we are a good way to being comfortable.
"But, if we hadn't won, the league starts to tighten up and you always have to be careful in the Premier League.
"Once you get complacent and setting your sights too high, it's got a habit of coming back to bite you. We won't do that."
Albion bounced back from successive injury-time defeats and Clarke admitted: "I'm happy. The performances were good in the last two games but we got no points.
"It was just important to get back on track. The performance was okay. It wasn't exceptional but okay.
"We had good control of the game, especially in the second half. We didn't look under too much threat and always looked like we could score on the counter-attack."