The government is planning to relax its travel quarantine rules in early July for some countries.
UK officials are talking to their counterparts in Portugal, France, Italy, Greece and Spain.
However, the UK hopes to make an announcement on 29 June that it has secured a number of "travel corridors".
The government had previously said that the quarantine would be reviewed every three weeks and 29 June marks the end of the first three-week period.
A travel corridor would mean that two people travelling in both directions between two countries would not have to self-isolate after they travel.
The first travel corridors could come into force on 4 July, although that date is by no means confirmed.
Even if certain routes are exempt, the travel quarantine for people arriving from other destinations will remain in place.
A senior aviation source has told the BBC that the quarantine could remain throughout the summer for anyone arriving from countries which do not have a travel corridor with the UK.
Portugal's foreign minister previously said that anyone in the UK thinking of going to Portugal this summer would be "most welcome" despite the coronavirus pandemic.
Augusto Santos Silva said he hoped an "air bridge" between the UK and Portugal could be secured by the end of June.
However, the broader travel quarantine is expected to remain in place.
What are the new rules?
People arriving in the UK should drive their own car to their destination, where possible, and once there they must not use public transport or taxis
Arrivals must not go to work, school, or public areas, or have visitors - except for essential support.
They are also not allowed to go out to buy food, or other essentials, where they can rely on others
Those arriving in England, Wales and Northern Ireland could face a fine of £1,000 if they fail to self-isolate for the full 14 days, while they face a £480 fine in Scotland. The maximum fine for repeat offenders in Scotland is £5,000.
Anyone arriving from the Republic of Ireland, the Channel Islands or the Isle of Man does not have to complete a form or enter quarantine upon arrival in the UK.
There are also exemptions for workers in some industries such as road haulage and medical professionals who are providing essential care.
The travel industry has been vocal in its criticism of the government's quarantine rules, warning that the isolation period will deter visitors and put jobs at risk. Some airlines were in the early stages of legal action.
The manufacturing industry has also highlighted that fewer flights will restrict imports and exports, which will have a knock-on effect for the freight industry, as well as hampering the recovery of some businesses.
Despite criticism from businesses, Home Secretary Priti Patel said that the measures were "proportionate" and being implemented "at the right time" when they came into effect on 8 June.