The midfielder, who missed Napoli's 2-0 win over Sampdoria on Monday, has been suffering with the injury for around a month.
He told the Swiss Football Association's official website that the problem had intensified in recent days and so he has decided to have an operation.
The Switzerland international added that he would be back in six weeks' time.
However, Napoli have moved to dismiss this, saying he will now undergo consultation to assess the injury and whether an operation is required.
"Napoli, in agreement with Valon Behrami, announces that the player in the coming days will undergo further consultation with a specialist of international repute to determine whether an operation on his foot is necessary or not," the Serie A side said in a statement on their official website.
Behrami has made 20 appearances for Rafael Benitez's side this season, but is yet to score for the club.
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