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.