Villas-Boas was shown the door by Tottenham on Monday just one day after they were hammered 5-0 by Liverpool at White Hart Lane.
The Portuguese is the fifth manager to leave a Premier League club this season and Allardyce's future has been called into question following West Ham's disappointing form.
The London club are just a point above the relegation zone after winning only three of their first 16 Premier League games.
But the former Blackburn Rovers boss is confident West Ham will start climbing the table when they welcome back a host of players from injury and suspension.
He said: "No, I don't fear for my job. 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 knows nothing is certain in football, but he is making plans to strengthen his squad in the transfer window next month and is hopeful he will be backed by the board.
"I'm not saying that it wouldn't happen (getting sacked), 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, which even if it didn't include (injured striker) Andy Carroll was a very good squad," he said.
"It's fraught with injuries and suspensions at the minute and that is a great difficulty for us to deal with. I worry more about my team than my own position.
"I always have done and I know everyone's feeling the pressure; it's always going to happen to someone. We all can't finish tenth, we all can't finish top and we've all got to find our place in this league and we've got to continue to strive to do that.
"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.
"We've got to go all out to try and get the squad back together, that's the main reason why we are where we are at the moment. We've got to make sure we do one thing better than we've done up to now."
