Migrants who arrived after the deal took effect Sunday were being led to previously open refugee camps on the islands of Lesbos and Chios and held in detention, authorities on the islands said.
EU countries are trying to avoid a repeat of the mass migration in 2015, when more than a million people entered the bloc.
Most were fleeing civil war in Syria and other conflicts, traveling first to Turkey and then to the nearby Greek islands in dinghies and small boats. Efforts to limit migration have run into multiple legal and practical obstacles.
"We are conscious of the difficulties," EU Commission spokesman Margaritis Schinas said in Brussels.
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Commission officials said support staff needed to implement the deal including hundreds of translators and migration officers" would not start arriving until next week. Returns, they said, cannot start until Greece changes its law to recognize Turkey as a "safe country" for asylum applications.