The Maranello outfit have so far failed to win a race this year after taking three victories in 2015.
"All I hope [for 2017] is that Ferrari get their act together and start winning races," Ecclestone told the official Formula 1 website.
"When I got Jean Todt to take his position and go to Maranello - which was a bit of a risk for Jean to do - it was an all-Italian team and they were a bit concerned about taking a foreigner."
"But I told them: when you win the championship you sure will find ancestry in Jean's family that comes from Sicily."
Ecclestone also said that he thinks Maurizio Arrivabene has a tough job and he doesn't envy him in the slightest.
"Now it has gone back to being a very Italian team again. And it is run like an Italian team."
"So I don't envy Maurizio's job. I wouldn't want to do it."
Bahrain International Circuit - Winter testing