The Kathimerini daily said the agreement came during a strained two-day state visit by Turkish President Recep Tayyip Erdogan this week, during which he angered his hosts with talk of revising borders and complaints about Greece's treatment of its Muslim minority.
The deal is in addition to Turkey's existing agreement to take back migrants from Aegean island camps, under the terms of an EU-Turkey pact.
Greek Prime Minister Alexis Tsipras' office did not respond to AFP requests for comment on the report. But a Greek government source yesterday said Athens and Ankara had agreed "new measures of cooperation towards decongesting the islands, under the terms of the EU-Turkey pact."
It is unclear how Brussels will react to the move, Kathimerini said.
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Criticised by rights groups, the EU-Turkey pact was designed to encourage Ankara to stem refugee flows.
Under that controversial deal, Ankara pledged to take back illegal migrants landing in the Greek islands in exchange for financial aid, eased EU visa rules for Turkish citizens, and limited direct resettlement of Syrian refugees living in Turkey.
The pact sharply reduced the number of migrants trying to cross the Aegean Sea to reach Greek islands, although Ankara has repeatedly threatened to walk away from the deal, including over the failure to fulfil the pledge on visa-free travel.