The Hammers have just won tenancy to the Olympic stadium.
Sullivan said: "If I was running Arsenal, instead of spending £10m each on three players, I'd spend £30-£40m on one.
"Arsene Wenger doesn't like spending money - it is a fact. For a club of our size we spend big money.
"If we spend £12-14m on a player this summer, in three years we may spend £30-40m. We have a number of targets but it's about spending wisely.
"Often you need a bit of luck. I'm seeing someone on Monday funnily enough that if it comes off will be the biggest signing in our history. We lack goals in the team, we have got to sign a top-class striker.