A coastguard official said they spotted a dingy transporting migrants towards Chios, which lies a few miles off the coast of Turkey, and ordered them to turn back.
"The smugglers maneuvered toward the coast guard vessel causing a collision," the official told Reuters.
The coast guard said 25 migrants were rescued but one of them, a woman, later died. A search and rescue operation was ongoing.
Two coast guard officers were injured and transferred to hospital, a second official told Reuters. Witnesses reported that about 30-35 people were on board, a government official said.
Greece, in the southeast corner of the European Union, has long been a favored gateway to Europe for migrants and refugees from the Middle East, Africa and Asia.
In 2015-2016, Greece was at the frontline of Europe's migration crisis and nearly one million people landed on its islands, including Chios, from Turkey.