Turkey, Ukraine and Kosovo should also expect more delays in visa waiving as the bloc turns more cautious amid immigration fears, EU delegation sources said, Censor.NET reports citing Reuters.
The EU is already making it easier to suspend visa-free travel before it grants such right to more states, most notably Turkey, whose help it needs to control immigration after some 1.3 million people reached Europe last year.
While Brussels says Turkey, with a population of 79 million, is making progress on 72 criteria to win the EU visa waiver, Ankara is seen missing an end-June deadline.
EU envoys in Brussels discussed a similar deal for Georgia again on Wednesday but there was no decision, with Germany and France among countries opposed, diplomats said.
They may return to the issue next week, before holding their first discussion on the more controversial case of Ukraine. More technical-level meetings on Ukraine are due only on June 14.
Immigration is also a key theme in the June 23 referendum in Britain on whether to leave the EU, struggling after 1.3 million refugees and migrants reached it last year.
The weakening political momentum also complicates the matter politically for the other three countries, diplomats say, with time running out before the summer break. The European Parliament, where a majority is needed to enact such agreements, will hold its last session on July 4-7 and only resume mid-September.
The visa waiver allows easier access - but not the right to work - for up to 90 days to Europe's free-travel Schengen area, which comprises most EU states and several non-EU ones.
About 2 million Kosovars, Georgians and Ukrainians hold the top-notch biometric passports that would make them eligible for visa-free travel, according to Brussels data. Turkey has issued no such documents so far.
That shows only a limited number of people would initially benefit. But those wary of visa liberalization say the EU cannot open up to 130 million people - roughly the combined population of the four states - as it faces migration and security risks.