"Amazon fulfillment centers processed and tendered customer orders to delivery carriers on time for holiday delivery," said Amazon. "We are reviewing the performance of the delivery carriers."
Walmart also will provide gift cards to customers who did not receive packages by the promised deadline, the New York Times reported.
UPS said the problems stemmed from a storm caused by an unexpected jump in Christmas shopping the last two weeks, a compressed holiday shopping season in 2013 due to the late date of Thanksgiving and some severe weather that halted some deliveries.
The delivery woes suggested the retail sector is still adjusting to shifting customer behavior with the rise of online shopping.
Amazon characterized its overall holiday shopping season as the "best ever" in the company's history.
Particularly popular was the "Amazon Prime" service, which provides free two-day shipping services and streaming of some television shows and movies for $79 a year.
Amazon said it signed up more than one million customers for its "Prime" service in the third week of December.