“I am right about Amazon costing the United States Post Office massive amounts of money for being their Delivery Boy,” Mr. Trump wrote in a tweet in April.

While the details behind the agreement are not public, available evidence suggests the opposite: that Amazon’s business has been a boon to the post office. Some current and former White House officials have said that Mr. Trump sometimes blends Amazon with The Washington Post, which has covered his administration aggressively. The Post is owned by Amazon’s founder, Jeff Bezos.

In mid-April, President Trump abruptly issued an executive order demanding an evaluation of the Postal Service’s finances. The order did not mention Amazon, but it was clear that he would like for the panel to substantiate his claims that the financial arrangement between the Postal Service and Amazon, its biggest shipper of packages, is a money loser.

Executives at Amazon dismissed any link between the new delivery program and Mr. Trump’s Twitter attacks about its agreement with the post office.

“This doesn’t play into that debate at all,” Dave Clark, Amazon’s senior vice president of worldwide operations, said in an interview Wednesday afternoon at the Admiral’s House, a historic landmark property overlooking Elliott Bay in Seattle.

Mr. Clark said his job was to think five to 15 years down the road about Amazon’s needs and that the new delivery program was designed to meet future growth and capacity demands. He added that Amazon would continue to use all of its partners, including the Postal Service, to get packages to its customers.