The mineral accounts for 10-15 percent of the South American country's GDP, with much of it exported to China, the world's biggest consumer.
Following the results of a vote that lasted until Saturday night, the union reported in a statement that there were 2,164 votes in favor of starting the strike against 11 for accepting the employer's offer.
Union and company leaders can initiate a final dialogue with government mediation within five to 10 days.
Escondida workers staged a 44-day strike in 2017, the longest in the history of Chilean mining. The strike caused US$740 million in losses for the company.
The workers are asking for a one-time bonus to recognize their work during the Covid-19 pandemic as well as education benefits for their children.
"We hope that this strong vote is the decisive wake-up call for BHP to initiate substantive talks... if it wants to avoid an extensive conflict, which could be the most costly in the country's union history," the union said.
The Escondida mine is located in the world's driest desert, the Atacama in northern Chile, at more than 3,000 meters (10,000 feet) in altitude.
It is the same area where in 2010 some 33 men were trapped 700 meters underground for 69 days following a cave-in at the Copiapo mine.
