The EU has named 14 countries whose citizens are deemed "safe" to be let in from 1 July, despite the pandemic - but the US, Brazil and China are excluded.
Those named include Australia, Canada, Japan, Morocco and South Korea, reports BBC.
The EU is ready to add China if the Chinese government offers a reciprocal deal for EU travellers, diplomats say.
EU border controls have been lifted for EU citizens travelling inside the bloc. Rules for UK travellers are covered separately in the Brexit negotiations.
UK nationals are still to be treated in the same way as EU citizens until the end of the Brexit transition period on 31 December. Therefore, during that time UK nationals and their family members are exempt from the temporary travel restriction.
On the current "safe" list, still likely to be amended, are Algeria, Australia, Canada, Georgia, Japan, Montenegro, Morocco, New Zealand, Rwanda, Serbia, South Korea, Thailand, Tunisia and Uruguay.
The UK is currently negotiating "air bridges" with several EU member states, so that coronavirus does not totally block summer holidays - the busiest season in Europe for tourism, which employs millions of people.
The EU procedure to formalise the list, and criteria by which countries are judged safe or not, are to be finalised by midday on Tuesday.
A qualified majority of EU countries - at least 55% of the EU countries, representing 65% of the EU population - have signed off on list.
There were splits between those such as Spain - wanting the boost of tourism, but preferring to play safe because they have been hit so hard by Covid-19 - and others like Greece and Portugal, which depend on tourism but are less scarred by the virus.
Last week reports said member states were assessing two different lists. The Politico website said one covered countries with fewer than 16 Covid-19 cases per 100,000 people and the other with up to 20 cases, which would include Canada and Turkey. The New York Times said the list would be revised every two weeks, so the US could be added later.
Earlier this month the European Commission also stressed that reopening borders with non-EU states in the Western Balkans was a priority from 1 July. However, EU member Croatia announced on Wednesday that travellers from Serbia, Kosovo, Bosnia and North Macedonia would all face 14-day self-isolation, because of an increase in infections.