The Teamsters Union said Wednesday United Parcel Service "walked away" from negotiations over a new contract, a claim that the parcel delivery firm denied amid growing chances of a strike.

"The Teamsters have stopped negotiating despite UPS's historic offer that builds on our industry-leading pay," the company said, adding that the union should return to negotiations.

The union had earlier in the day said UPS made an offer that was unanimously rejected and that the company had "walked away from the bargaining table."

The conflicting statements come as a contract covering full- and part-time employees of UPS in the United States that deliver, load trucks and handle packages expires at midnight on July 31.

UPS workers have already authorized a strike, should the talks break down.

Both the union and company officials have said before that they wanted a deal finalized to prevent a strike, which could put millions of daily deliveries at risk, including vital medicines for treating cancer and other illnesses.

The only national strike at UPS was in 1997. It lasted 15 days, disrupted the supply of goods, cost the company $850 million and sent some customers to rivals.

"UPS had a choice to make, and they have clearly chosen to go down the wrong road," said Sean O'Brien, general president at the Teamsters, which represents roughly 340,000 U.S. drivers, package handlers and loaders at the company.

UPS sweetened its offer last week, but O'Brien said it did not go far enough to reward workers who risked their lives to keep packages moving during the early days of the COVID-19 pandemic that fueled big profits for UPS.

The push for higher wages comes as labor unions are enjoying a higher bargaining power with companies, which have been grappling with labor shortages since the pandemic.

"Refusing to negotiate, especially when the finish line is in sight, creates significant unease among employees and customers and threatens to disrupt the U.S. economy," UPS said on Wednesday.

Shares of UPS were down 2.6% before the bell.