The West Ham United manager saw his team suffer a 3-1 defeat to Newcastle United on Saturday, leaving them in 18th place while Moyes’ Manchester United suffered a loss by the same scoreline away to Chelsea.
But while the spotlight on the Scotsman is undoubtedly strong at a club as big as the Red Devils, Allardyce feels the pressure on him is comparable.
He said: “People would think outside the pressure is probably greater on David than me but from my point of view I think it's an equal pressure - not only just now but also the future of this football club.
“The burden at this club is that it wants to move into the Olympic Stadium in two years’ time or so and wants to have a team in the Barclays Premier League.
“That stadium is obviously going to increase the quality. People will now think West Ham are going to be a big club by moving into this new big stadium. So you've got to build for that. That's a bit of extra pressure but that's a long way off.”
While Moyes has endured four defeats already in 2014 and has seen his side slip to 14 points behind league leaders Arsenal, Allardyce has also endured both cup humiliations and floundering league form.
The 5-0 defeat to Nottingham Forest in the FA Cup was compounded by the 6-0 debacle at Manchester City, and despite a vital 2-0 win away to Cardiff City, the pressure was back on Allardyce after the Newcastle defeat.
West Ham have now lost nine of their last 12 games, and, after tomorrow’s dead rubber clash in the second leg of the Capital One cup semi-final with City, face a visit to title-chasing Chelsea.
Allardyce has also had to deal with angry supporters with loud boos heard at Upton Park on Saturday.
The former Bolton, Blackburn and Newcastle manager said: “I think that all fans are disgruntled it their team isn’t doing well - whether it's West Ham or anyone else.
“The difference is how many fans are in the stadium. If things are not going so well then the boos are greater in a smaller stadium. So you get criticised when you are in this position.”
The Hammers have had to deal with a host of injuries to key players over recent months but Allardyce has positive news in terms of players returning to action.
He said: “Kevin Nolan and James Tomkins are back. Andy Carroll will get more minutes than in the last game. We hope he's getting closer to a full 90 minutes.
“Hopefully we can give Manchester City a really tough game. We will not have anyone of inexperience out on the field - apart from Ravel Morrison.”
The fixture with City may be largely irrelevant following the first leg result but Allardyce needs to get Carroll and co firing again if he is to ensure that the Olympic Stadium does indeed have Premier League football when West Ham arrive in 2016.
image: © Ben Sutherland