West Ham co-chairman David Sullivan confirmed five bids are in for new players as the club aims for signings with "good value'' in a "potty'' market.
But he stressed the club will not break the bank to try to bring in a player like Jamie Vardy.
England striker Vardy, 29, headed out on international duty at Euro 2016 in France with his club future the subject of speculation, following reports Arsenal have lodged a lucrative £22 million offer to attempt to lure him away from Premier League champions Leicester.
West Ham have already brought in Norway international Havard Nordtveit earlier in the summer, but have seen reported interest in Carlos Bacca, Alexandre Lacazette and Callum Wilson all fail to materialise.
While Sullivan hopes the east London side will soon be able to announce new arrivals as Slaven Bilic prepares a squad for life at the Olympic Stadium, he says those signings will not come at any cost.
"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,'' Sullivan told the West Ham official website.
"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.''
Sullivan added: "We are 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.''