Sky News political correspondent Rob Powell said “it is early days but the idea is being discussed in Westminster and people would instead have to take daily tests”.
The idea would only be brought in later in the year once the adult population has been offered both doses, Powell said.
He added that “this idea of offering more privileges to people who’ve had two jabs is clearly in the ascendancy”.
It comes after public health expert Linda Bauld said those who have had both doses of a COVID-19 vaccine may not have to self-isolate for 10 days in future.
She added that any change in regulations could see the UK following the US template – and the rule could stand even if they had come into contact with someone infected with the virus.
Professor Bauld, who is an expert in public health at the University of Edinburgh highlighted repeated statements by the UK government’s chief medical adviser Professor Chris Whitty and others that COVID-19 would not disappear and society would have to learn to live with it.