Ferrari boss Luca di Montezemolo reportedly called the driver on his 32nd birthday on Monday to tell him off for his negative comments.
The call came after Alonso was asked after the recent Hungarian championship what kind of car he wanted for his birthday.
"The one the others have," said the disgruntled driver.
When asked what he would do after over the one-month Formula One summer break, he said "I will pray," according to the UK's Guardian newspaper.
Ferrari were quick to respond, stating on their website that Alonso's comments "did not go down well with the president Luca di Montezemolo, nor with anyone in the team".
The racing team said: "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."
In the wake of Lewis Hamilton's dominant victory for Mercedes in Sunday's Hungarian Grand Prix, the Formula One paddock was abuzz with rumours that Alonso's manager had already held talks with Red Bull chief Christian Horner.
As the teams packed up ahead of the sport's four-week break, the better-established tradition of "silly season" speculation was ignited — and sent Ferrari into denial overdrive.
Ferrari claim that Alonso is under contract with them until 2016.
Both Horner and Alonso avoided direct statements to kill the story when they were confronted with it by reporters and Vettel himself talked openly about the prospect of partnering his Spanish rival.
Horner said: "Any conversations between any of the drivers and any of the parties will always remain confidential... But, of course, there have been quite a few drivers who have expressed an interest in the seat, as you can imagine."
Asked about the potential dynamics of an Alonso–Vettel partnership, he added: "It would be interesting. At the end of the day, you have to think about what is right for the team.
"Obviously the pairing that we want to put together for next year has to be right for the team. Last week, there was speculation about Kimi (Raikkonen). This week it is about Fernando.
"I think, for us, we want to make sure that nothing waivers in our mind of putting together the right team package for next year with the two fastest drivers that we can, who will work collectively well together and achieve the best results for the team."
The rumours began after Alonso's representative, Luis Garcia Abad, held a meeting with Horner at the Hungaroring on Friday.
It is believed they discussed the future of junior driver Carlos Sainz Jnr but also mentioned Alonso, who is known to be disappointed with Ferrari's inability to match Red Bull's pace
The Red Bull vacancy has arisen following Australian Mark Webber's decision to leave F1 for sportscar racing with Porsche next year. Webber and Alonso are also close friends.
"Is he (Alonso) available?" said Horner. "I don't know. You ask him."
Ferrari team principal Stefano Domenicali claimed that Alonso was locked into a contract with the Italian team but conceded that current performances did not impress him.
"First of all we have a contract with him," Domenicali said. "No doubt, Fernando is an asset for the team. We need him to work very closely with everyone, because after all this is a difficult moment in the season.
"We have to deliver a job. Full stop. This is the same for everyone, not only for him."
Alonso laughed when asked about a possible future at Red Bull and insisted he was "very happy" at Ferrari while Vettel said, if given a choice between Alonso or Raikkonen of Lotus, he would prefer the Finn.
"I'd prefer Kimi," he said. "I have to be careful now. Nothing against Fernando — I really respect him a lot as a driver.
"I respect Kimi on track and off track because he has always been really straight with me. From that point of view, it might be a bit easier."
Horner added: "There is quite a lot of interest...We'll take a bit of time over the summer to reflect on the options that are open to us and look to make the right decision to put the right driver alongside Sebastian before next year."