BRUSSELS—Hopes in Georgia, Ukraine and Kosovo that the European Union would approve their visa-free access to the bloc by the summer have faded in recent days amid a raft of new concerns raised by the bloc’s biggest member states.

The issues range from German concerns about organized crime by Georgian gangs to worries in France that the visa deals could create new security vulnerabilities at a time of heightened terror fears.

EU officials and diplomats now say that September appears to be the earliest date that EU governments and the European Parliament could sign off the agreements. The deal would give millions of citizens from these countries who hold biometric passports visa-free access to the border-free Schengen zone covering most EU nations. The U.K. and Ireland are outside Schengen.

For the bloc, the visa-free regimes are a key tool for binding their neighbors closer and for advancing reforms in those countries. Delays could deplete pro-western forces in countries like Georgia and Ukraine, who are struggling with their hostile Russian neighbor.

In recent months, the European Commission, the EU’s executive, has recommended visa-free status for Georgia, Ukraine, Kosovo and Turkey. The latter’s bid has already been put on the back burner with EU and Turkish officials now targeting an October deal as Ankara completes several final tasks.