The Potters have yet to add any new faces to manager Mark Hughes' squad for next season.
Coates says negotiations over potential targets has been "messy" because of the tournament in France, but is expecting some clarity soon.
"I'm sure things will start to move in the next few weeks," the 78-year-old told BBC Radio Stoke.
Stoke publically declared an interest in Portugal's European Championship-winning forward Nani, but has signed for Spanish club Valencia, and have also been linked with West Bromwich Albion striker Saido Berahino and Egypt winger Ramadan Sobhi.
"We're hoping to bring one or two new players in," Coates said.
"It's always messy with the Euros, as nothing gets done, but these things do take some time as clubs have their own agenda.
"You're not in control - you just have to be patient and hope things drop for you."
As Stoke wait to confirm their first new face, a familiar one - Glenn Whelan - has extended his stay for another year.
The 32-year-old Republic of Ireland midfielder, who is preparing for a his ninth full season with the Potters, has agreed a new deal until 2018.
"He's been a great player for Stoke City - if you have 11 Glenn Whelans, you'd be okay," Coates said.
"We have a great admiration for him, he's a real team player and he wants to win."
Stoke City chairman Peter Coates hopes the club can soon confirm some new signings now Euro 2016 is over.