Ambassadors from the 27 EU members convened from Friday afternoon to establish criteria for granting quarantine-free access from next Wednesday.
A redrawn text of 10-20 countries was put to them, but many said they needed to consult first with their governments, diplomats said. The list did not include the United States, Brazil or Russia, one diplomat said.
Discussions were continuing overnight, with the EU countries expected to give informal replies by Saturday evening, people familiar with the matter said.
U.S. passengers may be allowed to travel if they meet certain conditions such as passing temperature checks, two U.S. officials said.
The European Commission had advised that the bloc first lift internal border controls and then gradually open up to outsiders. However, the first step has not gone according to plan.
Greece is mandating COVID-19 tests for arrivals from a range of EU countries, including France, Italy, the Netherlands and Spain, with self-isolation until results are known.
The Czech Republic has said it will not allow in tourists from Portugal, Sweden and part of Poland.
There is broad agreement that the bloc should only open up to those with a similar or better epidemiological situation, but there are questions about how to assess a country's handling of the epidemic and the reliability of data.
A number of countries, such as Tanzania, Turkmenistan and Laos have no reported cases in the past two weeks, according to EU agency, the European Centre for Disease Prevention and Control (ECDC).
Based on ECDC data for the two weeks to Thursday, a range of countries are clearly in a worse situation than the European Union.
They include the United States, Mexico, Brazil and much of Latin America, Russia, South Africa and Saudi Arabia.
Despite pressure from U.S. airlines and unions, the White House has not committed to mandating fresh air travel safety measures in the wake of the pandemic. Discussions between airlines and government officials including Vice President Mike Pence on Friday over temperature checks ended without an agreement.
In a statement, Pence's office said the parties also discussed "the best path forward for allowing Americans to safely travel internationally again."
The Commission has suggested the western Balkans countries - Albania, Bosnia and Herzegovina, Kosovo, Montenegro, North Macedonia and Serbia - should be admitted.
However, according to the ECDC data, the number of cases in Bosnia and North Macedonia could be too high.