About 30 migrants were believed to be initially on board the vessel that foundered off the rocky shores of Agios Isidoros in the northwestern part of the Aegean island, according to coastguard officials.
The four migrants found dead were all women, a coastguard official said.
Five migrants have been rescued so far, among them a pregnant woman and a minor who were taken to hospital for medical checks, another official told Reuters.
The coastguard told Reuters another 25 migrants had been located on shore.
It was not immediately clear whether they had been on board of the vessel that sank.
Four vessels and a naval helicopter were involved in the search-and-rescue operation launched by the coastguard after a resident reported seeing migrants on the shore.
Strong winds in the area hindered the operation, the official added.
Greek authorities were also searching the island in case some of those believed missing had already reached the shore.
Greece was a favoured gateway to European Union countries for migrants and refugees from the Middle East, Africa and Asia in 2015-2016, when nearly a million people landed on its islands, mostly via inflatable dinghies.
The flow of people dropped off before resurging last year.
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