"What we want is for things to move quickly and for a June deadline to be chosen as the final deadline otherwise the withdrawal agreement will be more complicated," Le Drian told reporters in Dublin before a meeting with his Irish counterpart Simon Coveney.
With Ireland remaining in the European Union but Northern Ireland being part of the United Kingdom, which is set to leave in March 2019, the question of the border has become a vexed question in London, Brussels and Dublin.
The British government has said there will be no "hard border", which is generally agreed to be necessary to maintain the Good Friday Agreement, which put an end to the Northern Ireland conflict.
But Prime Minister Theresa May has pledged to be free of the EU customs union so that it can reach its own trade deals with the rest of the world.
Her Conservative Party is split over what form of customs agreement should be negotiated with the bloc.
The EU has proposed a "backstop proposal" in which Northern Ireland would remain in the customs union without the rest of Britain.
The EU has given a mandate to negotiator Michel Barnier "which includes two major elements - no recreation of a physical border and the Good Friday agreement", Le Drian said. "A backstop proposal exists. If the British have another, they should tell the negotiator."
Coveney said he did not believe the question could be resolved through technology and that political agreement has to be reached.