Amazon could face an investigation by the Advertising Standards Authority over complaints that its premium service is failing to deliver on time in the run-up to Christmas.

Amazon Prime claims to offer “unlimited one-day delivery” but customers have contacted the advertising regulator to say it is falling short of what is promised.

@AmazonUK 3 orders via Prime in a week all 3 delivered late and 1 is missing! Amazon have a rule of customers having to wait 3 days for an investigation to check with the courier where the order is. Won't bother renewing Prime — Mandy Kaur (@mandy_malhi) December 16, 2017

A spokeswoman for the Advertising Standards Authority (ASA) said: “We have received a handful of complaints. We are considering whether to launch an investigation.”

Customers of the online retailer can choose free delivery above a minimum spend but at the checkout are offered a free 30-day trial of its express delivery service, which costs £7.99 a month thereafter.

Amazon says the last order date for standard delivery is Wednesday but for Prime, which it says offers “amazing delivery benefits on your Christmas shopping”, the last order date is Saturday.

Information elsewhere on the site says: “Your order will be dispatched with the intention that it’s delivered one day after dispatch.” It advises customers to contact customer services if they do not receive a parcel by the estimated delivery date.

The consumer rights group Which? said: “If you paid for delivery by a certain date or time (eg by Christmas or next-day delivery) and the delivery arrives late, this is a breach of contract. If it was essential that your goods were delivered on time, you have the right to terminate the purchase and get a full refund.”

In 2015 the ASA investigated Amazon after six people claimed advertising for Prime was misleading. It upheld complaints that the advert in question, a direct mailing sent to existing Amazon customers, did not make sufficiently clear that a paid subscription would automatically start if not cancelled during the free trial and did not state what the cost of the subscription would be. The regulator concluded that the ad was “likely to mislead”.

The Amazon UK press office could not be contacted for a comment. The Sunday Telegraph quoted an Amazon spokesman as saying: “We’d ask any customers with questions about their deliveries to contact us.”