The warning issued Thursday came as nearly a third of the 52,000 migrants stranded in Greece by European border closures are refusing to move to organized shelters and remain camped out at the port of Piraeus and at the northern border with Macedonia.
More than 4,000 migrants remain at Piraeus - an important site for Greece's vital tourism industry.
"Every effort will be exhausted to persuade refugees and immigrants that it is in their own interest for them to move," a statement from the Greek coast guard reported by the Associated Press said. "There is a 10-15 timeframe for them to leave the port."
Athens has toughened its position toward migrants since a March 20 agreement between the European Union and Turkey went into effect. Some 4,000 migrants and refugees who reached the Greek islands from Turkey after that date are in detention, with most due to be sent back to Turkish ports. The deportations started Monday and are expected to resume Friday.
More than a million refugees and migrants reached the EU last year, most traveling through Greece and across the Balkans to central Europe.
At Piraeus on Thursday, Interior Ministry officials and translators spent hours trying to persuade migrants to move to a new camp in northern Greece, showing them aerial photographs of the site and explaining what facilities are available.
"We are trying to explain that the new camps have good facilities and that people there will be able to fill out their asylum applications there," volunteer translator Ilias Iakovou told the AP. "But people are afraid to go because they fear they will be cut off and will run out of money. They feel safe if they are near Athens."