Official travel advice against all but essential travel outside Britain will also be eased for some countries and regions.
Taken together, these changes will make it easier for Britons to travel abroad for summer holidays.
The relaxations are the latest taken by the government to unwind emergency measures put in place to stop the spread of COVID-19, as ministers look to limit the economic damage caused by the virus.
An expert panel will put nations into three categories: green, amber and red. Passengers arriving from green and amber countries will no longer have to quarantine themselves for 14 days after their arrival.
"Our new risk-assessment system will enable us to carefully open a number of safe travel routes around the world," a government spokeswoman said. "But we will not hesitate to put on the brakes if any risks re-emerge."
The rules for red-category countries will not change.
The quarantine policy, introduced on June 8, has been heavily criticised by airlines, airports and the hospitality sector who say it deters international travel at a time when they had been hoping for it to recover.
The categories are due to be announced next week, with the rule changes expected to come into effect the week after. They will be based on factors including prevalence of COVID-19, the trajectory of the disease and the reliability of data.
Britain said it is likely to discuss this with countries including France, Greece and Spain.
The Foreign Office will next week announce those countries where it considers the public health risk is no longer unacceptably high.