"We watch what they do," Abney said on "Squawk on the Street." "They're certainly a customer, but they can be a competitor in certain areas, and we monitor that."
Big-time logistics companies such as UPS are in a precarious situation with Amazon as the e-commerce giant builds out its own delivery business.
Amazon represents about 12% of UPS' revenues, according to CNBC's Frank Holland, and Abney noted the two companies' relationship grew, on a percentage basis, in the last year.
Abney attributed the growth to three things: competitive winds, structural change that Amazon and others instigated around next-day delivery, and a muted international supply chain due to macro challenges such as trade tensions.
"We've got a mutually beneficial relationship," Abney said, suggesting the current relationship between the companies wouldn't be disrupted "as long as it stays mutually beneficial."
Shares of UPS were down more than 5% on Thursday after the Atlanta-based company reported earnings.
UPS saw quarterly revenue growth of 3.6%, but it was still slightly below Wall Street estimates. UPS also issued a full-year earnings-per-share forecast below expectations.
One day earlier, UPS announced it placed an order for 10,000 electric vehicles from U.K.-based start-up Arrival, a move that came months after Amazon announced it planned to buy 100,000 electric vehicles from Rivian.
UPS and Amazon also are both taking part in a New York City pilot program designed to reduce road congestion by making deliveries on cargo bikes. DHL is the third company in the pilot.
FedEx has taken a significantly different approach from that of UPS, announcing in August that it would not renew its ground-delivery contract with Amazon.
The change is part of the company's efforts to "focus on the broader e-commerce market," a FedEx spokesperson told CNBC at the time.
The rivalry between the companies heated up around the holidays, when Amazon temporarily prohibited third-party sellers from using FedEx's ground-delivery shipping.
Amazon began allowing the third-party sellers to use the service again on Jan. 14, sending shares of FedEx almost 2% higher on the announcement.
Shares of FedEx are down about 18% over the last year. UPS' stock is up about 8% over the same period.