The Frenchman had only joined Bordeaux on Monday after spending two seasons with Serie A outfit AC Milan.
He entered the pre-season friendly against Lorient in the 62nd minute but had to be taken off 15 minutes later after losing part of his right ear.
Menez was inadvertently stepped on by Lorient's Didier Ndong, leading to the horrific injury.
A Bordeaux statement read: "Jeremy Menez had part of an ear cut off during the encounter against Lorient, which led to his premature exit in the 77th minute."
Ndong was remorseful following the gruesome incident, apologising to Menez via a Lorient statement.
"I offer my apologies to Jeremy Menez and Girondins de Bordeaux following a totally unintentional play which had unfortunate consequences for the Bordeaux striker," the 22-year-old said.
"I give him my full support for the coming days and hope to see him very quickly on the pitches of Ligue 1."
"A daily update will be made with the surgeon in the coming days," a Bordeaux statement also reads.
"The downtime of Jeremy Menez will be dependent on the healing. We wish our player a speedy recovery and we hope to see him on the pitch soon."