Richard filed a complaint at a Paris court last week before heading to Israel in a bid to calm the controversy, a source close to the affair disclosed.
"Stephane Richard received death threats on his telephone and his personal data were published on an Internet site in the context of the controversy over Orange's presence in Israel," the source said.
Once these personal details were published, Richard received hundreds of calls, including death threats, added the source, who did not wish to be named.
The French telecoms giant declined to comment on the issue.
The row erupted after remarks by Richard during a news conference in Cairo on June 3 in which he said the firm was planning to withdraw its brand from Israel at the earliest possible opportunity.
Orange licenses its name and logo for use by Israeli firm Partner Communications under an agreement due to run until 2025.
Israel reacted furiously to Richard's comments, accusing him of bowing to a Palestinian-led boycott campaign.
His comments were interpreted in Israel as a response to a report by a group of NGOs which accused Orange of indirectly supporting settlement activity through its agreement with Partner.
Even Prime Minister Benjamin Netanyahu became involved, calling on Paris to renounce "the miserable action of a company that is under its partial ownership".
Richard flew to Israel last week in a bid to smooth ruffled feathers and told Netanyahu he "deeply" regretted the controversy and would never support any boycott of Israel.