The Hammers boss and his side begin their Cup campaign with a Third Round tie against Everton on Tuesday night.
Everton v West Ham United
The FA Cup
Third Round Proper
7.45pm, Tuesday 6 January 2015
Goodison Park, Everton
Live on BT Sport
Winners receive £67,500 from prize fund
And after an impressive start to the season in the Premier League which sees them sat in seventh spot at the turn of the year, Allardyce believes his side are ideally placed to launch a Cup run.
"Everybody wants to get to Wembley, everybody wants to go and play in a Cup Final," said the former Bolton Wanderers and Blackburn chief.
"I think we're in a position where we can hopefully go for it, but it's a hugely difficult tie getting past Everton away."
The current mood around the Boleyn Ground could not be more different to what it was 12 months ago, when Allardyce’s side were fighting against relegation from the top flight.
And when they were thrashed by 5-0 against Nottingham Forest in the Third Round Proper last January, Allardyce admits it was one of his lowest points in management.
But the spirit in the Hammers squad is now buoyant and the team are relishing each and every game according to their boss.
"Everybody's looking forward to coming in to training and you can sense the camaraderie and atmosphere among the players is really good," added Allardyce.
"It [the spirit] never got bad here, but it did get subdued this time last year because of the poor results we were getting."
And that is something which long-serving midfielder Mark Noble fully agrees with.
The homegrown star has been a regular throughout Allardyce’s spell in east London and reckons the morale is as good now as it has been in his time with the club.
"It's a great place to be at the minute, after last year when it was a horrible place to be," revealed the 27-year-old.
"It will be fantastic to go on a Cup run. Last year, it would have been tough as we didn't have the players to cope with it.
"But now, the manager can pick a stronger side and we can hopefully progress."
Watch the full video preview with both Allardyce and Noble in the media player below.
West Ham boss Sam Allardyce and midfielder Mark Noble speak about the possibility of an FA Cup run