The online retailer said that only a "small number" of incomplete orders were sent out, but has supplied all customers with a code, meaning many now have two copies of the title.
"It's been brought to our attention that a small number of orders for this item may have been delivered without the advertised game, Nintendo Land, due to supplier error," read a statement from Amazon.
"In the event that your order was one of the small number missing the game, please use the following code and instructions below, to download the game."
It has not been confirmed whether customers who ordered the console from elsewhere were affected by the issue, or whether other retailers are offering a similar compensation scheme.
The Wii U was released in Europe on November 30 following a successful launch in North America.
> Wii U: Everything you need to know about Nintendo's new console
> Wii U: Almost half of Digital Spy readers not interested in new console
Watch Digital Spy go hands-on with the Wii U below: