Elkann spoke before being given a prize at an event in Washington DC to mark the 50th anniversary of the National Italian American Foundation.
Vasseur, who has been in his role since January 2023, was given a new contract by Ferrari two and a half months ago.
Hamilton said on Thursday at the United States Grand Prix in Austin that the stories about Horner and Ferrari were "distracting".
Elkann's comments came after Ferrari issued guidance implying there was no truth to the Horner claims.
In Saturday's sprint race in Austin, Hamilton finished fourth with team-mate Charles Leclerc fifth.
