DETROIT - The United Auto Workers' strike against General Motors could continue for the foreseeable future, as the union on Sunday said talks between the sides have "taken a turn for the worse."

UAW Vice President Terry Dittes, in a letter to members on Sunday, said the union plans to continue to negotiate but a Sunday morning proposal from GM was inadequate and "reverted back to their last rejected proposal and made little change." The offer, he said, was a counter proposal from an "extensive package" the union provided to the automaker on Saturday night.

"The Company's response did nothing to advance a whole host of issues that are important to you and your families!" Dittes wrote. "It did nothing to provide job security during the term of the Agreement."

GM, in response to the letter, said the company continues "to negotiate in good faith with very good proposals that benefit employees today and builds a stronger future for all of us. We are committed to continuing discussions around the clock to reach a resolution."

Both sides declined to comment on details of the discussions, however a source familiar with the talks told CNBC that one of the union's main concerns with the most recent offer was the company's lack of commitment to new products for U.S. plants. Prior to the talks, GM was expected to attempt to negotiate the closure of up to four domestic plants, including large assembly plants in Ohio and Michigan, to address overcapacity.

The talks, according to the source, have "cooled considerably" from the progress that was being made in recent days, including Friday, when the union publicly said "good progress" was being made on several issues. While the UAW's rejection of GM's most recent offer could be a setback, the talks continue and neither side has walked away from the table —a positive sign for the negotiations overall.

Dittes, in a separate Sunday letter released by the union to GM, said the company "didn't even have a professional courtesy" to explain why the UAW's most recent proposal was rejected: "We expect the Company to respond and discuss the package proposal we presented yesterday," he wrote. "The law and basic decency require no less."