Cole will become a free agent at the end of the month when his contract with Premier League champions Chelsea expires, while former Arsenal striker Henry could leave Spanish giants Barcelona this summer.
Since taking over West Ham in January, Gold and co-chairman David Sullivan have made it clear they are intent on making a "marquee" signing that would boost the stature and commercial backing of the club.
After failing in their quest to sign former Manchester United striker Ruud van Nistelrooy, the duo turned their attention to Henry, and then Cole, who began his career at the club before moving to Chelsea for £6million in 2003.
The club have made offers to Cole and Henry and expect to hear back from both players after the World Cup.
"We, as much as anyone, want England to do well at the World Cup so there's absolutely no chance of us talking to Joe during the World Cup," Gold said.
"We wouldn't do that during this very delicate period but after that, that's a different matter.
"When it comes to big transfers we don't like to any stone unturned.
"The chance of getting Cole is 10/1 and therefore there is a one in 10 chance of us getting him. We probably won't get him but the offer is still there.
"There is the same chance of Henry coming but we will try.
"We had a chat with him for a bit but it looks like he is going to be offered another contract.
"The offer is still on the table for him too and it will be there for another two or three weeks.
"We have had it in the past where we have thrown the net out far and we will throw the net out again in the future.
"When we were the owners of Birmingham City, nobody would have thought that we would have been able to get Christophe Dugarry but we did and he had an amazing half a season and he kept us in the Premier League."
The duo are at pains to emphasise they will not jeopardise the club's financial future by over-spending but bringing in either player would undoubtedly boost ticket sales and give the club a bigger presence commercially.
One person who will not be leaving the club, according to Gold, is England goalkeeper Robert Green.
"Robert Green has two years left on his contract and we hope he will start the season and be with us for years to come so we are very happy with him," Gold said.
"He is a world class goalkeeper and we want him to stay with us."
Gold and Sullivan chose to replace Gianfranco Zola with Avram Grant after the Italian narrowly avoided relegation to the Championship last season.
The Israeli, who guided Chelsea to the Champions League final in 2008 and led Portsmouth to the FA Cup final last term, has promised to bring an attacking brand of football to the club and has already added midfielder Thomas Hitzlsperger to his squad.
Gold believes Grant, 55, is capable of taking the team up the table, with a place in Europe, he feels, not beyond the club's reach.
"Europe is a possibility," said Gold. "Avram is just starting work though. The team we had last season virtually got us relegated so we'll have to improve it to get there.
"We were very strong in certain areas and we were very weak in others. You can't have that.
"He has already identified some positions that he would like strengthened and we will be looking at players that we can present to him and he will be advising us on players who he wants as well."