The England forward is reportedly nearing a move to north London after Arsene Wenger’s men activated a release clause in his contract believed to be worth around £20m-£22m.
The 29-year-old fired Leicester City to the Premier League title last season, firing 27 goals for club and country.
The Hammers have been linked with a move for Vardy, but Sullivan says a deal would have broken their ‘current wage structure’.
Providing fans with an update on the club’s official website, Sullivan wrote: “On the pitch, work continues to secure the targets we have identified to ensure we build on the considerable progress made in Slaven Bilic’s first season as manager here.
“It was reported that one of those targets was Jamie Vardy, and while it is true that we would loved to have had him, the numbers being talked about in a potential move to Arsenal show that the market is going potty.
“It is safe to say that signing him would have smashed our current wage structure and we are determined to buy players with good value in this mad market.
“We have done that exceptionally well in recent years, with the likes of Winston Reid, Cheikhou Kouyate, Aaron Cresswell, Dimitri Payet and Manuel Lanzini vindicating this approach.
“All of those players have gone on to become stars of our team and you can rest assured that we will be bringing players in this summer who will prove to be just as influential.”
With Euro 2016 around the corner, few deals have been complete since the end of the Premier League season, and Sullivan revealed United have ‘five bids’ in place.
He added: “We’re working 24 hours a day, seven days a week to make this happen, and with five bids in for players at the moment, we are hopeful that we can bring you some exciting news soon.”