Port police responded to a distress call, sending three boats and a tugboat to help the stricken vessel which came from Turkey, the coastguard told AFP.
According to the coastguard statement, the tugboat reached the sailing boat carrying the migrants but did not pull it to safety because it was taking on water and there were high winds.
The migrants were initially taken to a rocky islet nearby with other boats and were transported to Mykonos early Sunday.
Search and rescue efforts were continuing.
Migration minister Notis Mitarachi had earlier tweeted that eight migrants were missing and 104 had been rescued.
He also called on Turkey to do more to protect human lives and eradicate smuggling networks.
"Greece is protecting its borders with determination, but its first priority is always the protection of human life," shipping minister Giannis Plakiotakis said.
In May, Greek authorities said they had prevented around 600 migrants from crossing the Aegean Sea from neighbouring Turkey in one day, in the largest attempted entry this year.
A migration ministry source had said migration flows to the Greek islands in the first four months of 2022 were nearly 30 per cent higher than in the same period last year.