The 27-year-old England international is currently in Switzerland with the Hammers squad for a pre-season training camp but is set to hold talks with Stoke boss Tony Pulis upon his return.
Potters chairman Peter Coates told Sky Sports News: "We have agreed terms with West Ham, that's done. Now it's a case of can we agree terms with Carlton?"
Cole joined West Ham from Chelsea in July 2006 and has scored 42 goals in 165 appearances, but he found the net just five times in the Barclays Premier League last season and is now poised to move on in the wake of the Hammers' relegation.
The fee is believed to be an initial £4million with a possible £2million add-ons.
Coates admits this is not the first occasion Stoke have attempted to sign Cole but is hoping this time a move will come to fruition.
"We come from two different places. He has got an injury problem - he has always had that - and that is a risk you take so we have to factor that in," he said. "But he is a good player and Tony has always liked him. Manager's like different players but Tony has always liked him.
"I think we've tried to sign him around three times over the past few years without succeeding so maybe we will get somewhere this year."
Cole is currently away in Switzerland as part of the Hammers' 26-man squad, which Coates admits has delayed a potential deal.
"We're still on the case," he added. "The team is still away at the moment. They come back Thursday this week, I think. Tony likes to be around for these things."
Source: PA
Source: PA