The Hammers endured an inconsistent 2013-14 campaign as they laboured to 13th, but co-chairmen David Sullivan and David Gold believe Allardyce is the right man to have them challenging for European football this season.
Sullivan has lofty ambitions for the club, insisting they should be aiming for the top six rather than a solid mid-table finish.
Back in May, he said: "I don't want to just aim for the top 10. We should be aiming for the top six. I want to be looking down the table not up it next season."
But Allardyce has tempered expectations of a first top-six finish since 1999, and he believes repeating of their top-half finish two years ago is the limit of what the club can hope for.
"We're trying to build a team that gets where we want to go," he said.
"Let's not forget that year one was a massive overachievement which built expectation. Finishing 10th in your first year (after promotion) is a major success, and there's a backlash to that success always in the game - not just at West Ham, but wherever you are.
"Do fans think you should finish 11th? No, they think you should finish ninth, eighth or seventh. You've always got to live with the expectation but we have to stay real and say that if we have a very good season, we can finish better than 10th.
"We'll be around 10th or 11th in the Premier League I think. That's somewhere we want to try and achieve; it's year three for us - it's not year one or two - we should be looking more to establish ourselves rather than hopefully worrying about relegation."
Allardyce has been backed in the transfer market this summer and has brought in seven new faces, including big deals for Enner Valencia and Cheikhou Kouyate.
But the 59-year-old reckons budget and financial clout are the main factors as to what a club can realistically aim to achieve.
'Across the board, you generally finish around where your wage budget lies," he said.
"If you've got the biggest wage budget, then you're more than likely to finish in and around the top four, generally. And then you've got your 10th or above, then you've got below 10th and then you've got the bottom end.
"Over the last five or 10 years, most teams finish around where their wage budget is. There are obviously some anomalies in that - some teams one season can have a particularly good season and some teams have a particularly poor season."
Sullivan's statement could be seen to put pressure on Allardyce's position if the Hammers do not make an early impression, but Allardyce added: "In terms of what the chairmen and the fans expect, all that's fine.
"We can all say that and I want the same, but the reality is we know how to get there - whether we'll get there is another factor on various anomalies as we go through the season.
"For instance, this week - and it's only game two - we may only have one centre-half fit and that's the last thing you want when you're going into a new season when you're talking about the same problems you had last December. Already we're feeling the pinches on the value of the team and the selection process that make or break your season."