The fears came to light after it was revealed that more than a quarter of children own a handheld computer before they reach the age of eight.
Ernest Doku, a telecoms expert from Uswitch.com, the online comparison website which published the report, said: "Today's kids know their way round the latest tech before they're out of nappies."
Overall, the research estimates around 2.1million children under the age of eight, equal to 27 per cent of children in this age group, have a tablet, such as Kindle, used for reading books.
It said nearly one in three children learned to use a table or a smartphone 'by the time they were three'. One in ten were under the age of two.
Researchers interviewed 571 parents with children under the age of 16 about their offspring's hi-tech habits.
When asked if they thought their children aged up to 16 were 'addicted' to gadgets, 16 per cent of parents said their children were obsessed by technology.
Mr Doku said: "Parents really do need to keep tabs on what their children get up to online and lay out some ground rules.
"If not, they risk having to cover the cost of bills racked up by in-app purchases [buying things while using an application, typically a game, on their handheld computer]."
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