"All the great champions who have driven for Ferrari have always been asked to put the interests of the team above their own," the team website quoted president Luca di Montezemolo as telling the Spaniard in a telephone conversation to wish him a happy 32nd birthday.
"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," he added.
Alonso finished fifth at the Hungaroring and slipped to third in the championship, 39 points adrift of Red Bull's triple champion Sebastian Vettel with nine races left.
Asked afterwards what kind of car he wanted for his birthday, the double world champion told Italian reporters: "The one the others have.”
Asked also what he planned to do over the August break, with the season now at its midpoint, the Spaniard had retorted: "I will pray".
In a blunt assessment, the Ferrari website said Alonso's comments "did not go down well with Montezemolo, nor with anyone in the team" and said the president had "tweaked" the driver's ear in their conversation.
The polemic came after Alonso's manager was seen entering the Red Bull motorhome in Hungary, triggering paddock speculation that the champions could be considering the Ferrari driver as a possible team-mate for Vettel.
Alonso has a contract with Ferrari to the end of 2016 and is the team's undisputed number one driver, with Montezemolo usually unstinting in his praise.
Ferrari had earlier announced a reshuffle in their engineering department with James Allison arriving from Lotus as technical director from September 1.
Pat Fry will assume a new position as director of engineering and both men will report to team principal Stefano Domenicali.
Allison, who worked closely with German driver Michael Schumacher in a previous stint with Ferrari, left his last job as Lotus technical director in May.
Ferrari said Allison's arrival was proof of Montezemolo's determination to ensure the team had all the support and resources it needed.
"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," the website said.
It added that the president had made his displeasure with the team's current performance clear.
"Montezemolo did not mince his words when it came to asking the team to step up a gear," the website declared.
"Each one of the engineers present received a 'gift' of a knife, along with an invitation - metaphorical up to a point - to put it between their teeth when thinking how to tackle the second half of the season."
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