In March, Luxottica announced a partnership with Google to develop and distribution fashionable versions of Google Glass, the smart glasses device.
Luxottica declined to comment on the press speculation but a spokesperson said the company was considering its strategy for the future.
"We can confirm that for some time the Chairman Leonardo Del Vecchio and the CEO Andrea Guerra have been debating the best strategic direction for the group," a spokesperson told CNBC via email.
Guerra took the helm of Luxottica in 2004 and has been credited with a massive turnaround of the company. Under Guerra's leadership, the company's share price has more than tripled.
Several Italian newspapers reported that Guerra could leave at the end of the year when his contract expires, but the prospect of his departure has investors worried.
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"It does leave much room for nervousness," Cedric Rossi, luxury and consumer good analyst at Bryan Garnier, told CNBC in a phone interview.
"Considering the outstanding performance of Guerra at the helm I am not surprised by the negative reaction in the stock."
Luxottica posted strong first-half results last month, reporting net sales of 3.9 billion euros - up 5.6 percent year-on-year.