Di Montezemolo has been chairman of Ferrari - with focus on both racing and road cars - for more than 20 years, and has enjoyed significant success on and off the track.
However Ferrari have failed to win a World Championship since they took the constructors' crown in 2008, leading to rumours that he will exit the company in the near future.
And Sergio Marchionne, the head of FIAT Chrysler, seemed to confirm that over the Italian Grand Prix weekend, criticising the team's performance at their home race. Kimi Raikkonen came home ninth, more than a minute shy of race-winner Lewis Hamilton, while Fernando Alonso retired with an engine problem.
“The heart of Ferrari is winning in Formula 1," said Marchionne. "I don’t want to see our drivers in 7th and 12th place. To see the Reds in this state, having the best drivers, exceptional facilities, engineers who are really good, to see all that and then to consider that we have not won since 2008...
"The important thing for Ferrari is not just the financial results, but also winning and we have been struggling for six years."
The comments came after Di Montezemolo had remarked that he will choose if and when he leaves Ferrari, with Marchionne hinting that the decision may be made for him.
“On volume and economic results Luca has done an outstanding job. We are good friends but when I read his statements, these are things I would not have said myself.
“I consider myself essential, of course. But I also know very well that I am at the service of this company. So to create positions, illusions that one can operate outside the rules, is talking rubbish.
“It’s the same for him as it is for me, we serve the company. When the company has a change of plan, or if there is no longer a convergence of ideas, things change.”
Di Montezemolo has been linked with a new role at Italy's national airline Alitalia should he leave Ferrari.