Border staff will no longer have to check if arrivals have completed a passenger locator form if the computer system fails to find one, The Guardian reported on Tuesday.
A memo seen by the newspaper confirmed electronic border gates "will no longer refer passengers to in-person checks by Border Force officers if a passenger locator form is not found".
A passenger locator form is used by authorities to trace people who may have come into contact with someone with Covid.
The memo, said to be sent to staff on July 19, also reportedly suggested passengers will not have to provide proof of a negative Covid test before they fly, or confirmation they have booked a follow-up test after they are back in the country.
The ISU - the Union for Borders, Immigration and Customs - has since apparently confirmed the Guardian’s reports.
Lucy Moreton, of the ISU trade union, confirmed the reported memo was accurate to the BBC.
"Ultimately this is a political decision," Ms Moreton said.
"Certainly, it will reduce queue times significantly and hopefully also the level of verbal abuse to which Border Force staff are subject.
"That is welcome to us. The impact on the UK’s Covid security is ultimately a scientific determination."
Last week, images posted on social media showed long lines of travellers at Heathrow Terminal 5.
A spokeswoman for the airport said a number of staff had been forced to self-isolate.
“Earlier today we experienced some passenger congestion in Terminal 5 departures, due to colleagues being instructed to self-isolate by NHS Test and Trace.
“We have activated additional team members to assist passengers with their journeys and the operation has now returned to normal.
“We apologise to our passengers for any inconvenience caused.”
The Liberal Democrat chair of the all-party parliamentary group on coronavirus, Layla Moran, told The Guardian: “This beggars belief. Scrapping Covid checks at our borders will make it far harder to detect and keep out variants from abroad.
“It seems the government is intent on repeating the mistakes that allowed the Delta variant to become dominant in the UK.”
The Home Office said said airlines were still legally required to conduct a series of Covid checks.
A government spokesperson said: “Our utmost priority is protecting the health of the public and our enhanced borders regime is helping reduce the risk of new variants being transmitted.
“All passenger locator forms are still being checked by carriers, as they are legally required to do, and to suggest otherwise is wrong. This legal requirement on carriers is underpinned by a robust compliance regime, which is overseen by regulators.”
The spokesperson added: “Compliance with these rules is essential in order to protect the population from new variants of Covid-19, and so there will be tough fines for those who do not follow the rules.”
Airlines UK, which represents easyJet, Tui and British Airways among others, told the BBC: “Carriers are committed to working with Border Force on these requirements, which is why compliance has remained consistently extremely high.”