The transaction still requires approval from Cooks' shareholders at a special meeting next month, but should it go ahead the brothers will become the firm's largest shareholders, with a stake of more than 50 per cent.
Esquires was founded in Canada in 1993, where another firm holds the rights, and the Deeks established the business in Australasia before selling the Australia and New Zealand franchise rights to ASX-listed Retail Food Group for $11.6 million in 2011.
The brothers retained the rights to the rest of the world including Asia, the Middle East, North Africa and the Pacific Islands.
The company earns a royalty and supplies the franchisees with coffee and other products made in New Zealand.
Deeks said Esquires was already operating 11 stores in China and its Chinese franchisee, a state-owned firm linked to the Government of Yunnan province, is planning to open 150 stores in the country over the next three years.
"It's a bit like having Auckland Council being your franchisee," he said.
Deeks said the coffee trade was taking off in the Middle East, where Esquires operates one store in Bahrain - where that country's royal family is the franchisee - two sites in the United Arab Emirates and nine stores in Saudi Arabia.
"Retail is really strong in that part of the world," he said. "It's how people entertain themselves, particularly in places like Saudi where there's no movie theatres, bars or nightclubs."
Deeks said the first Esquires store in Kuwait would open this week and his firm had just signed a "letter of intent" with a franchisee in Turkey who intends to open 50 stores.
The aversion many Kiwi and Australian consumers had to coffee chains did not exist in other parts of the world, he said.
He added that Esquires' New Zealand ownership gave it a boost with consumers in international markets, particularly Saudi Arabia.
Cooks chairman Keith Jackson said the company - which was also in negotiations to buy a Bay of Plenty-based food processing firm, as well as some other retail businesses - was attracted to Esquires' established brand and growth potential.