The West Ham manager believes Potters boss Tony Pulis will be unhappy with his side so far this season having won just twice in the Premier League, making them more dangerous
Stoke have won just twice so far this season and lie in a lowly 14th place, while the Hammers have acquitted themselves excellently after triumphing in the Championship play-offs last season, and are currently seventh - six points ahead of the Potters.
Pulis has earned plaudits for keeping Stoke in the English top flight since their promotion in 2008, and Allardyce has praise for the way the Welshman has forged a successful side at the Britannia Stadium.
“Tony Pulis' team are well-drilled and disciplined in the way they want to play," the 58-year-old told reporters. "I think that they get the defensive side of it right, as I try to do as well.
"I think the improvement in the quality of their players has made them an established Premier League side now.
"I think Tony probably thinks they are under-achieving a bit at the minute with the team he has built over the last few years. He had a more difficult start to the season in terms of fixtures than we did, so I'm sure they'll get back to winning ways, but I hope it's not on Monday."
Allardyce joined the Irons after their relegation from the Premier League in 2011, and guided the club back to the top flight in his first season in charge.
He believes an important aspect of that success was his relationship with chairmen David Gold and David Sullivan as well as chief executive Karen Brady.
Tony Pulis is known to have a good understanding with the Coates family who own Stoke, and Allardyce thinks that is a key part of the Potters’ Premier League success.
"I think it's true that Tony Pulis has had a great relationship with his owner and that relationship has borne fruit, particularly because they have backed him and know that they can trust him,” he added.
"I think that trust has given him the finance to build year in, year out and that has made them what they are today. They have improved every year in terms of the quality of player that they have brought in.
"I think that's why they've been successful, along with the team spirit they have and how organised they are. That relationship, buying wisely and maximising resources has borne fruit for Stoke City."
Pundits are beginning to recognise the excellent start West Ham have made to the season, but Allardyce is not getting carried away with his side’s form.
He said: "The tough phase has already begun for us and, after Stoke on Monday we go into a run against the other big boys. We've coped very well so far and I hope we continue to cope on Monday night because that's a big game for us.
"We want to continue our home form and home results and get a very important three points against a well-experienced and well-organised team with a really good manager."