Tunisian authorities denied a drone was involved and said they were investigating a fire in a life jacket aboard the boat.
The Global Sumud Flotilla said a vessel known as the "Family Boat", travelling under a Portuguese flag, "was struck by a drone", but that no one aboard was hurt.
Video posted on the group's social media showed a flash of light landing on one of the boats and setting off what appears to be a fire.
The group, whose flotilla was scheduled to leave Tunisia on Wednesday, is expected to give a news conference in Tunis later on Tuesday about what happened.
It said the incident would not derail its mission to deliver aid to Gaza.
The flotilla is part of a broad movement seeking to deliver aid to Gaza by boat.
The Family Boat was carrying the most famous members of the flotilla, including Swedish activist Greta Thunberg and former Barcelona mayor Ada Colau.
It is still unclear who was on board during the reported attack.
The Tunisian interior ministry said reports about a drone attack were unfounded, and that specialised security units are investigating the results of a fire in a life jacket.
The Israeli military and Israeli government did not immediately respond to requests for comment.
The flotilla - which activists call "Steadfastness" in Arabic - set sail from Barcelona last week hoping to transport food, water and medicine to Gaza in defiance of Israel's blockade on the enclave's maritime border.
The group had speculated about drones flying overhead on livestreams and social media.
Activists argue that the presence of doctors, artists, clergy and European politicians on board will help spotlight the blockade and Israel's moves to enforce it.
Its voyage comes three months after a smaller activist flotilla crossed the Mediterranean with plans to deliver aid to Gaza.
An overland convoy travelling across North Africa also attempted to reach the border but was blocked by security forces aligned with Egypt in eastern Libya.
Israel has previously dismissed such efforts as publicity stunts, saying a blockade is necessary to prevent smuggling, particularly of weapons.
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