Cole slipped out of the Premier League with West Ham last season and looks certain among those heading out of the club this summer.
Stoke are long-time admirers of the England forward and have had a bid accepted by the Hammers - believed to be an initial £4million with a possible £2million in add-ons.
Coates is hopeful the Potters will now finally get their man after failing on a number of previous occasions in recent years.
"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 with West Ham, while the Stoke squad is in Austria, and Coates says no deal will be done until Pulis meets the striker himself.
"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.
"Now we've done a deal with the club it really needs his input with the player because he likes to talk to them and make sure they are on board, and he is on board and everything is ok from his point of view.
"But we see that work in progress and hope that something can be done."
QPR were also thought to be monitoring Cole's availability but he could now be set for a move to the Britannia Stadium.