The first fatalities since the Turkish government opened its border again last week allowing Syrian to migrants reach Europe have been reported.
A child died after being pulled from the sea when a boat capsized off the Greek island of Lesbos.
While Greek officials have denied claims by two Turkish sources to Reuters that migrants were being prevented from crossing into the country Monday (March 2), which lead to the death of a one refugee.
At least 1,000 migrants have reached Greece's Eastern Aegean islands since Sunday morning Greek police say.
More than 10,000 migrants are believed to have attempted to cross by land at the border, where guards from both sides fired tear gas into crowds caught between the fences in no-man's land.
Turkey is currently home to 3.7 million Syrian refugees and has another million on its doorstep from a new surge of fighting, said last week it would stop enforcing a 2016 agreement that had prevented migrants from reaching the EU.
Greek government spokesman Stelios Petsas accused Turkey of orchestrating a coordinated effort to drive migrants across the frontier.
More than 130 people have been arrested trying to cross the border since Saturday, Greek police sources said.
The coast guard said the boat which capsized off Lesbos on Monday (March 2) morning had been escorted there by a Turkish vessel.
And Greek ministers have been quoted calling it an invasion and has promised to protect its borders.