Ministers are finalising plans for a series of "travel corridors" that will mean people arriving into the UK will not need to self-isolate.
Participating countries could include France, Italy, Spain, Greece, Belgium, Germany, Norway, the Netherlands, Turkey and Finland - but not Portugal.
Restrictions for these countries are expected to be lifted in early July.
The first travel corridors could come into force on 4 July, although that date is by no means confirmed.
Sweden is also unlikely to be on the list because the infection rate there is higher than in the UK.
Transport Secretary Grant Shapps had indicated there would be an announcement on Monday.
But the BBC understands that initial details could be unveiled this weekend.