Negotiations between the delivery company and the union representing 340,000 of its workers have been at a standstill for more than a week with a July 31 deadline for a new contract approaching fast.

The union has threatened a strike if a deal is not reached by the time the collective bargaining agreement expires. Asked during a webcast with members Sunday on whether the White House could force a contract on the union, Teamsters President Sean O’Brien said he has asked the White House on numerous occasions to stay away.

“My neighborhood where I grew up in Boston, if two people had a disagreement and you had nothing to do with it – you just kept walking,” O’Brien said.

“We don’t need anybody getting involved in this fight,” he said.

The Teamsters represent more than half of the Atlanta-based company’s workforce in the largest private-sector contract in North America. If a strike does happen, it would be the first since a 15-day walkout by 185,000 workers crippled the company a quarter century ago.

Before contract talks broke down, both sides had reached tentative agreements on several issues, including installing air conditioning in more trucks and getting rid of a two-tier wage system for drivers who work weekends and earn less money. A sticking point in negotiations is wage increases for part-time workers, who make a minimum of $16.20 an hour, according to UPS.

Last week, UPS said it will temporarily begin training nonunion employees in the U.S. to step in should there be a strike.