The Old Lady will host Mauricio Pochettino's Spurs in the first-leg of their Champions League Round of 16 tie on Tuesday night.
"The match against Tottenham will be very challenging," said Volpi.
"The way they played in the group stages should warn to keep the guard high."
Pochettino is attempting to take Spurs into the final eight of the competition for just the second time.
Over the past three-and-a-half seasons, the Argentine has managed to establish the north London club amongst the top clubs in England.
And Volpi is confident the 45-year-old, who has been linked with Real Madrid, could replicate his success in Italy.
He added: "Pochettino is a great manager and I have no doubts he would do very well in Italy too."