The UK Advertising Standards Authority (ASA) is considering whether to launch a formal inquiry into Amazon.
The ASA said: “We have received a handful of complaints about Amazon parcel deliveries and we are at the initial assessment stage.”
An Amazon spokesman told the BBC the ASA had confirmed to it there was no investigation at this time.
The ASA said there had been five complaints since the start of the month.
Amazon offers free delivery above a certain minimum spend, but express, unlimited delivery for Prime customers.
It also offers this coming Saturday, 23 December, as its final order date for Christmas orders, three days later than the Wednesday offered for non-Prime customers.
The BBC has been inundated with comments from Amazon Prime customers. Most reported problems with deliveries.
Gary from Farnham in Surrey, said: “Our Amazon Prime deliveries often miss their ‘next day’ target. We always make a point of complaining, and are generally given a 30-day extension to our annual subscription.”
Craig from Carluke in South Lanarkshire, said his Prime orders were rarely delivered within a day: “They offer next-day delivery but it actually takes two days to come, which also isn’t part of the agreement which I pay for. Not acceptable.”
But other customers were happy. One pointed out that it was “unreasonable” to expect perfection every time. “It is a fantastic service and it is totally unreasonable to expect that things may rarely not work out,” he said.
Amazon advises customers to contact customer services if they do not receive a parcel by the estimated delivery date.
The consumer rights group Which? points out that a company missing its delivery date is in breach of contract.
It says that consumers have the right to terminate the purchase and get a full refund in that instance.
It says the first step should be to contact the retailer directly to make a complaint that a parcel has not been delivered and request they take action to remedy the situation.