"This subject cannot be dealt with in the coming weeks and months. That is not the wish of France," Cazeneuve said when meeting with his EU colleagues in Brussels.
The European Commission has urged member states to lift visa requirements on the three countries, as they have met all the bloc's benchmarks.
To take effect the proposal requires the approval of the European Parliament.
German Interior Minister Thomas De Maiziere said during Friday’s meeting that no new visa liberalization decisions should be taken before the snap-back mechanism is in place.
Georgia and Kosovo are also hoping to be granted visa-free access to Europe in the coming months.
The French ambassador to Russia, Jean-Maurice Ripert, earlier said that France was ready to lift visa restrictions with Russia as there was no risk of any big migrant flows from this country.
But France cannot do this alone as the move “requires the approval of all members of the Schengen Protocol,” the ambassador added.