The union said the company's latest proposal would have included two plant closings.
"The company left us with no option," said Ron Hoover, executive vice president of the United Steelworkers of America, which represents the Goodyear workers. "We cannot allow additional plant closures after the sacrifices we made three years ago to help this company survive."
The old contract expired July 22 and both sides agreed to an indefinite day-to-day extension. The Steelworkers issued a 72-hour notice on Monday and said they would terminate the contract at midday Thursday if an agreement wasn't reached.
The company said it was prepared to keep its plants open and take care of its customers but did not immediately say how it planned to do that.
Goodyear ranks No. 3 in the world in tire sales, based on revenues, behind top-ranked Bridgestone and No. 2 Michelin, according to the trade publication Tire Business.
Goodyear spokesman Ed Markey declined to comment on whether the company's offer involved plant closings.
"Our final offer to the union included a plan to secure retiree medical benefits and provide job security and investment guarantees for USW plants," Markey said. "The union rejected a comprehensive set of proposals that mirror the other industry agreements."
The union said it represents 15,000 employees at 12 Goodyear plants in the U.S. and four plants in Ontario, Canada. By the company's count, the U.S. plants have about 12,600 employees represented by the Steelworkers.
In Akron, where the 108-year-old company is based, workers at Goodyear's research center streamed out of the building at 1 p.m. and joined union members holding signs on the picket line.
About 20 pickets showed up outside Goodyear's sprawling headquarters carrying placards that read "USW Local 2L on strike against Goodyear Tire & Rubber for unfair labor practices."
Within minutes, scores more arrived to join the picket line, where people chanted. Goodyear office workers coming out at lunch time walked past the pickets without stopping and motorists honked horns as they passed.