West Ham have bids for five players currently on the table, but have been priced out of a move for Jamie Vardy, according to the club's co-chairman David Sullivan.
Sullivan told Sky Sports News HQ in May he was confident of securing a "top class forward" ahead of the club's move to the Olympic Stadium and confirmed on Monday West Ham were interested in signing the Leicester and England forward.
Sky sources understand Vardy has held talks with Arsenal over a move to the Emirates Stadium after the Gunners triggered a release clause in his contract with a £20m bid.
Vardy, who flew out with the rest of the England squad to France on Monday morning, is still yet to decide on his club future ahead of Euro 2016.
"It was reported that one of our targets was Jamie Vardy, and while it is true that we would have loved to have had him, the numbers being talked about in a potential move to Arsenal show that the market is going potty," Sullivan told the club's website.
"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."
West Ham have already had bids for Carlos Bacca, Alexandre Lacazette and Callum Wilson turned down, while Sky sources understand the Hammers remain interested in Michy Batshuayi and Christian Benteke.
Sullivan also reassured the club's supporters that he is working hard to secure new signings, revealing bids were currently on the table for five players.
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."
Norway international Havard Nordtveit became West Ham's first signing of the close season in May and Sullivan is hopeful any further new additions can prove as successful as some of his previous acquisitions.
"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," Sullivan said.
"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."