On the day of his 32nd birthday, Di Montezemolo on Monday telephoned the Spaniard to pass on his congratulations.
However, Ferrari have also confirmed Di Montezemolo "tweaked the ear" of Alonso, reminding him of his responsibilities to the Scuderia.
Ferrari have not made clear what Alonso is supposed to have uttered to have incurred Di Montezemolo's anger.
But after finishing fifth at the Hungaroring, in one interview Alonso is understood to have been asked what he would like as a birthday present, to which he replied: "Another car!"
Alonso is understood to be frustrated he again finds himself hunting Sebastian Vettel in the title race, falling 39 points behind the German who was third in Hungary heading into the summer break.
Via a news article on their website, Ferrari said: "There is a need to close ranks, without giving in to rash outbursts that, while understandable in the immediate aftermath of a bad result, are no use to anyone.
"That was a reference to the latest comments from Fernando Alonso, which did not go down well with Montezemolo, nor with anyone in the team.
"So, when Montezemolo called the Spaniard this morning to wish him a happy birthday, he also tweaked his ear."
Di Montezemolo is then quoted as saying he reminded Alonso that "all the great champions who have driven for Ferrari have always been asked to put the interests of the team above their own.
"This is the moment to stay calm, avoid polemics and show humility and determination in making one's own contribution, standing alongside the team and its people both at the track and outside it."
No mention was made of whether Di Montezemolo also asked Alonso about the rumours that circulated over the weekend linking him to Red Bull.
Alonso's manager, Luis Garcia Abad, spoke with Christian Horner, the Red Bull team principal, over the weekend, supposedly with regard to a rising star on his books in Carlos Sainz Jnr.
Horner, however, on Monday had numerous opportunities to deny Alonso was also on the agenda as he said: "We're in a fortunate position where there is quite a lot of interest in the seat for next year.
"Any conversations with other drivers or other parties are always going to remain confidential."
Alonso, however, still has three years remaining on his deal with Ferrari beyond this season, so is unlikely to make any move to Red Bull given the finances required to buy out his contract.
This is the avoidance of polemics Di Montezemolo speaks of as he requires his star driver to focus on the job at hand rather than look elsewhere.
Horner appears to be playing a crafty game because by refusing to rule Alonso out of the running, he knows it could unsettle the man he sees as the biggest threat to Vettel's hopes of a fourth successive title this season.
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