After knocking out AC Milan this week, Feyenoord will now meet Inter in the round of 16.
Zanetti said, "Our journey in the Champions League has been excellent so far, but now another competition begins. Feyenoord is a team to be respected, we have seen what they have done. We will prepare for the two games as best we can, there are all the conditions to move forward but with caution.
"We always have a great responsibility, we are Inter and this is an important competition. We will try to fly the Italian flag as high as possible.
"We can't make choices (Champions League or Serie A), we have to aim to get to the end of both competitions. We want to win the championship and then there's the Champions League, we are the last Italian team to have won it and we will try to honour it to the maximum by trying to get to the final like two years ago.
"It's difficult, but our mind and our thought is to get to the end of the two competitions."