Footage 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 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 Tunisian Interior Ministry said in a statement that reports about a drone attack were unfounded, and that specialized 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.