Contract negotiations between the United Parcel Service and the International Brotherhood of Teamsters, the union that represents UPS workers, broke down last week.

If the two sides do not agree to a new contract by the end of the month, UPS workers could go on strike for the first time since 1997.

The Teamsters, which represent more than 340,000 delivery drivers and warehouse workers for UPS, say the company made more than $100 billion last year, but still pays part-time workers a starting salary of $15.15 per hour.

"We just want a piece of the large pie that UPS is feasting on," Sean O'Brien, the general president of the International Brotherhood of Teamsters, told Errol Louis Tuesday night on "Inside City Hall."

The union is demanding vehicles with air conditioning, the end of a two-tier pay system, plus pay raises for full-time and part-time workers, who O'Brien says "make UPS the success that they are."

"There's an unsung hero within UPS. And those are the 100,000 part-timers that are well under $20 per hour," O'Brien said.

A spokesman for UPS says the Teamsters are "refusing to negotiate" and have a "responsibility to remain at the table."

"The Teamsters have stopped negotiating despite historic proposals that build on our industry-leading pay. We have nearly a month left to negotiate. We have not walked away," a UPS spokesperson said in a statement.

However, O'Brien calls that "fiction at best," saying UPS walked away from the negotiating table on July 5 at 4:15 a.m.

"[UPS] is going to choose whether they strike themselves," O'Brien said. "We've been transparent. We've been open and honest for a year."

O'Brien, a fourth-generation and 33-year Teamster member, says that this movement is not just about UPS, but also "to improve the lives of every single person that is in this arena."

He added that the Teamsters are looking to organize Amazon workers, as well.