Europe's internal borders began reopening to travellers on 15 June.
EU ambassadors meet on Wednesday to plan reopening external borders on 1 July, and travellers from the US could be among those not allowed in.
A number of European countries are keen to open up to tourists but others are wary of the continued spread of coronavirus.
The 27-member bloc must first agree the measures that non-EU countries should meet before deciding on a safe list.
The virus is spreading in the US, so it is likely Americans would be barred.
Brazil, Russia and other countries with high infection rates would also be left off a safe list, according to reports from Brussels.
The EU is not yet thought to have agreed how they will assess which countries meet health standards - one of the criteria for entry. Part of the problem is assessing reliable health data, reports say. Read more >>