The two children had boarded a dinghy in Turkey with their families to set sail for Kos, whose northern coast lies just 250 metres away, port police said.
One of the children's bodies has been recovered, but the search for the second victim, a six-year-old boy, is still under way.
His father, who was among 14 people rescued from the sticken vessel, had told rescuers that he had been unable to save his son from drowning and had had to let his body go in order to help other passengers.
More than 700,000 people have made the perilous journey across the Mediterranean this year so far and some 3,440 people have died or gone missing trying to do so, according to UN figures.
Greek Prime Minister Alexis Tspiras today paid a visit to one of the beaches on the island of Lesbos, which has become a prime gateway into Europe for the desperate refugees.
"A crime is committed in the Aegean (Sea) which must be stopped," Tsipras said afterwards, accompanied by European Parliament chief Martin Schulz.
Other protestors, wearing orange life jackets symbolising the risky exodus towards Europe, gathered at the so-called hotspot at Moria where new arrivals are registered and was also visited by Tsipras and Schulz.
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