Italian news media have speculated that Ferrari are interested in offering former Red Bull boss Christian Horner, dismissed in July after two decades of success, a return to Formula One.
Elkann said: "I want to express our full confidence in our Team Principal, Fred Vasseur, and in the work he is carrying out together with all our colleagues at Scuderia Ferrari - the mechanics, engineers, and drivers competing this weekend in Austin.
"I also want to reaffirm the importance of team-work by everyone to maintain focus on the only goal that matters: always giving our very best on the track."
His statement came just two days after Ferrari's drivers, seven-time champion Lewis Hamilton and Charles Leclerc, had fielded questions during news briefings on Thursday ahead of this weekend's United States Grand Prix.
"I mean, I don't know whether I can really shed much light on that, but it's a little bit distracting for us as a team," Hamilton said.
"Obviously, the team have made it clear where they stand in terms of re-signing Fred, and Fred and I and the whole team are working really hard on the future for the team. So, these things, naturally, aren't helpful."
Leclerc said: "There are lots of speculations around me, but just in general around the team for whatever reason. I feel like there are too many people speaking things not coming from actual facts and it's just a little bit annoying.
"It's always been like that... And I think, as a driver, but I think as a team, we've just got to focus on our things, which is what we are doing."
Ferrari are without a win this year and Hamilton is yet to claim his first podium finish for the team that last won the constructors' title in 2008 and the drivers' in 2007.