A humiliating 3-1 defeat at home to Independiente on Monday left River in 16th place in the Apertura championship with five points from eight matches.
It was the first time in 13 years that River had lost at home to Independiente, and 11th overall in one of the Argentine game's big 'classics'. Independiente climbed to third, one point behind leaders Estudiantes and title holders Velez Sarsfield.
"We know at what time your wife goes shopping" and "we know your family's movements" were among the threats made, they reported.
"We don't know (officially) what happened," he told Reuters. "The matter was not even discussed at tonight's board meeting."
River, one of the country's two most popular clubs along with their rivals Boca Juniors, are not spending on players while the club builds up to elections for a new chairman and board near the end of the year.
The Independiente match was the first of coach Leonardo Astrada's second spell in charge after he took over last week from Nestor Gorosito, who resigned following the previous weekend's 2-1 loss to San Lorenzo.
Big striker Cristian "Ogre" Fabbiani bore the brunt of fans' wrath after being substituted during the second half and made matters worse when he asked them to shut up by putting a finger to his lips. He later apologised.
