Senior cabinet ministers are due to discuss the move for arrivals to England in a meeting on Wednesday.
Senior cabinet ministers are to discuss allowing fully vaccinated travellers from the EU and US to avoid quarantine when they arrive in England.
A review of the rules is due by 31 July – the second date in the Department for Transport’s plan for a safe return to international travel.
Sources said the isolation exemption was likely to be discussed at the Covid Operations meeting on Wednesday.
But they said a decision on whether to proceed will not necessarily be taken.
Downing Street and the Department for Transport declined to comment on newspaper reports the government would go ahead with the plan.
But the aviation industry has been pushing for a relaxation of quarantine rules for travellers from the EU and US after completing a trial of checking the vaccination status of passengers.
British Airways, Virgin Atlantic and Heathrow Airport wanted to demonstrate that vaccination status could be checked away from the border and allow safe entry to the UK from countries on the amber list.
The companies said 99% of documents were verified correctly during the 10-day trial, which involved about 250 fully-vaccinated participants from the US, the Caribbean and Europe, travelling to Heathrow.
Two passengers had their credentials rejected, the companies said: one because their vaccination was completed less than 14 days before travel, and the other because of a discrepancy between the name on the passport and on the vaccine card.
Under current rules, other countries are granted a “traffic light” status for arrivals – red, amber or green.
The vast majority of countries, including the US and many European countries including Spain, Italy and Germany, are on the amber list.
Adults who have been fully vaccinated in the UK, and under-18s, no longer have to self-isolate after visiting any amber country apart from France.
But anyone who was fully vaccinated outside the UK still has to quarantine for 10 days on arrival, or pay for the test-to-release scheme to shorten their quarantine.
The travel industry has criticised the “frustrating” traffic light system for hindering its recovery. British Airways, Virgin Atlantic and Heathrow Airport said the UK was falling behind the EU in opening up to international travellers.
BA chief executive Sean Doyle said the trial provides the evidence that the government needs to allow fully vaccinated visitors from low-risk countries to come to the UK without self-isolating.
“The UK needs to safely reopen its borders as soon as possible to ensure loved ones can reunite, business can thrive and global Britain is able to take advantage of the UK’s world-leading vaccination programme,” he said.
Virgin Atlantic boss Shai Weiss said the UK’s current “overly cautious approach” would harm its economic recovery and put half a million jobs at risk.
Heathrow boss John Holland-Kaye said: “The vaccine has been a miracle of science, and these trials have shown that we can allow fully vaccinated passengers from the EU and US to visit the UK without quarantine.”
He said there was “no reason to delay with rolling out the solution from July 31”.