The Barcelona player is said to have confronted Javier Gomez Matallanas, who writes for Spanish newspaper AS, in the mixed zone after leaving the Arena da Baixada following Monday's 3-0 win over Australia.
According to The Guardian, while walking through the mixed zone, Alba said to the reporter: "Next time I'll rip your head off."
After Matallanas expressed his shock, Alba continued: "Yes, yes, next time ... I'm warning you."
The Barcelona star is believed to have continued his exchange with Matallanas while the team boarded a flight back to their homeland from Brazil, accusing the reporter of being a 'grass'.
Matallanas told radio station Onda Cero: "I have no idea why. I have not written anything about him. I don't remember writing anything about him at all."
The reigning champions were booted out of the World Cup after losing their first two group games of the tournament.