Earlier at Monza, even though the Red Bull switch rumours have ended, F1 chief executive Bernie Ecclestone let slip he thinks the Spaniard is "not happy" at Ferrari.
The Italian team has been playing down reports of tension between Alonso and Luca di Montezemolo, but the Ferrari president was visibly furious after qualifying at Monza -- and refusing to give interviews.
Alonso had complained on the radio that teammate Felipe Massa did not cooperate in an attempt to give the 'number 1' a high-speed slipstream.
He qualified fifth, behind Massa, and reportedly slammed the team on the radio afterwards. According to former Ferrari driver Jean Alesi, the Spaniard told his team members they are "stupid".
Another report quoted him as saying in Italian: "You have to let him (Massa) go!"
Although he put on a brave face when speaking to reporters afterwards, insisting it had been "a good qualifying", it is possible Alonso's dream relationship with Ferrari is over.
Earlier on Saturday, McLaren's Jenson Button was quoted by Brazil's Totalrace: "I heard an interesting rumour today.
"Someone told me about Fernando going to McLaren.
"I don't know if it will happen -- he didn't exactly leave McLaren in a good way (in 2007)," he added. "Let's wait and see."
After Saturday's events, Eddie Jordan also smells a rat.
"Fernando's not a happy boy," the former F1 team owner said on British television BBC.
"We could be watching something unfold here that we were not expecting."