Erick Thohir says the Nerazzurri must allow their captain to choose his next step but is open to prolonging the Argentine's stay at San Siro.
Chelsea boss Jose Mourinho - who knows the Argentine from his time in charge of the Nerazzurri - is reportedly prepared to offer the player a one-year deal that would then lead to a permanent coaching role.
Thohir revealed his respect for the club captain and said the decision is entirely down to Zanetti.
“I met the captain two months ago, it’s his decision,” Thohir told reporters. “He is a legend and we will respect any decision he makes. If he wants to continue or become a director then we will discuss it in the coming months."
The 40-year-old, who signed from Banfield in 1995, was promoted to club captain in 1999 and boasts five Scudetto titles and a Champions League medal amongst a sizeable haul of silverware.
Zanetti is also Argentina's most-capped player in history with 145 international appearances.