Ecclestone, 84, told the official formula1.com website at the weekend's Monaco Grand Prix that team bosses needed to be more prominent.
When it was put to him that Arrivabene was "quite a character", he replied: "But only for himself and not for Formula One".
Arrivabene, a marketing man who previously worked for team sponsor Philip Morris and took over his high-profile new job last November, said his responsibilities were to Ferrari.
"With all due respect, I don’t have to respond to Bernie for what I do, but rather to Mr Marchionne," said the Italian, referring to the Ferrari chairman and Fiat Chrysler chief executive, in a comment to Reuters.
"If Bernie has doubts about my way of doing, he should ask the team."
Ferrari's fortunes have been transformed since Arrivabene took over as the third principal in the space of a year at Maranello.
Sebastian Vettel was second in Sunday's Monaco showcase race, his fifth podium finish in six races since he joined from Red Bull, and Ferrari are now established as the closest rivals to champions Mercedes.
Last year they failed to win a race, their first blank season for 21 years.
(Reporting by Alan Baldwin, editing by Justin Palmer)
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