The Portuguese was sacked by Tottenham - who the Hammers face in the Capital One Cup this week - on Monday to become the fifth manager to leave a top-flight club this term.
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"No, I don't fear for my job," he told reporters. "When you're a senior manager in the years that I've been managing now, then that becomes one of those things that you face on some occasions of your career.
"There's always going to be a time when there's a difficult period and my responsibility is to manage the players through that. You have to make sure you bring stability back to the club and get where you want to be.
"My particular situation will ease and will get better as long as we get our inured players fit, and our communication between the board of directors and myself with everyone at the club is clear and precise."
Allardyce is keen to bolster his squad in January as West Ham look to improve the one-point gap between themselves and the relegation zone, but the boss is also aware that his crop have been hit by injuries this term - namely record-signing Andy Carroll.
He continued: "I'm not saying that [getting sacked] wouldn't happen, but we're supportive in terms of trying to do something in January to help the players and try to bring back the injured players to get the squad back to where it was at the start of the season.
"I don't have to prove a point to anybody. The only thing I do is continue to do my job and all the outside press can do and say what they want to because that's their right and their industry.
"My confidence is in myself and how I run a football club and I've been doing that for many, many years now. We're in a difficult time and a difficult period, I know why."