More than 500 workers at the site in Wakefield are angry at a pay offer worth 2.5%, the Unite union said.
The union warned that if its members took industrial action it could have a "devastating effect" on supplies to supermarkets and pubs across the UK.
Regional officer Kelvin Mawer said he was seeking talks with the firm.
'Below inflation'
Mr Mawer said: "Coca-Cola Enterprises (CCE) now face the prospect of a strike which we believe will have a significant impact on supplies of Coca-Cola during the hottest months of the year.
"Wakefield is one of the most productive and cost-effective sites in the UK. Yet CCE are effectively proposing a pay cut in real terms.
"How can a business like CCE expect its employees to accept a pay offer far below the rate of inflation when the same employees have contributed to the success of the business?"
A spokesman for Coca-Cola said it would be "inappropriate" to comment while negotiations with the union were ongoing.
But he said: "Decisions made by Coca-Cola Enterprises on employees' pay and rewards reflect our responsibility to protect the future vitality of our business for the collective benefit of the company and our employees."
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