"The message for everybody is to follow the rules," European Migration Commissioner Dimitris Avramopoulos told a news conference held with Luxembourg Foreign Minister Jean Asselborn, whose country holds the rotating EU presidency.
Many refugees from Syria, Iraq and Eritrea are currently failing to lodge asylum requests upon arriving in Greece because they fear they will be trapped in the recession-hit country.
However, this is a necessary requirement to entitle them for relocation under the EU scheme to share out 160,000 refugees within the bloc.
Avramopoulos and Asselborn had earlier visited registration facilities on Lesbos, the Greek island on the front line of the migrant influx that has overwhelmed Europe this year.
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Greece also plans to open registration hotspots on the islands of Samos, Leros, Kos and Chios, supervised by a command centre at the main port of Piraeus near Athens.
The EU's aim is to have the centres ready by the end of November.
But what is going to happen to thousands of non-Syrian migrants who will inevitably be sent back to Greece by other EU states is less clear.
The UN refugee agency on Friday said some of its staff had to be briefly evacuated from the registration site of Moria on Thursday, a day before Avramopoulos and Asselborn visited the facility as part of their visit to Lesbos.