Greece and Austria are urging other EU states to adopt coronavirus vaccination "passports" which could help revive Europe's stricken tourist industry.
The idea of such a document, likely to be a certificate, would be to permit those who have been vaccinated to travel freely within the EU.
The proposal was put forward during a virtual discussion between EU leaders.
But a vaccine passport faces opposition from some of the bloc's 27 member states.
France and Germany say such documents could be premature because data on the efficacy of vaccines in preventing a person from carrying or passing on the virus is incomplete.
There are also concerns that enabling a vaccinated minority to enjoy foreign travel while others, such as young people who are not seen as a priority for inoculation, continue to face restrictions would be discriminatory.
A further complication is the rapid spread of more contagious Covid variants - the English, South African and Brazilian forms - and the possibility of future mutations. So it is more likely that people will need booster jabs to remain protected.
Greece - as well as Israel - already has digital vaccination certificates, and others such as Denmark and Sweden have talked about developing them.
Greek Deputy Prime Minister Akis Skertsos told the BBC that a common digital certificate "is not discriminatory at all". He argued that non-vaccinated tourists could also visit Greece this summer, but the procedure for them would be slower - they would have to be tested and might have to self-isolate on arrival.
Greece and Cyprus have agreed to admit Covid-negative Israeli tourists this summer - those who can prove their status with the Israeli "green" digital certificate.
