The San Siro outfit missed out on Europe's premier competition this season after a poor 2013-14 campaign that saw them finish eighth.
Milan have shown signs of improvement under the stewardship of coach Filippo Inzaghi this term, but a 1-0 loss at Genoa last weekend has left the club seventh and five points adrift of third place.
Nesta - who won two Serie A titles and two Champions League medals during a 10-year stay - feels the club must perform better on a week-to-week basis in order to push for a Champions League return.
"I have travelled the world and the name of Milan is always too important to finish eighth," the centre-back told the Gazzetta dello Sport.
"They can make the Champions League. Obviously before they played to win it and now they play just to get in it.
"[They can only qualify] if they show more consistency. Inter, Napoli and the others are also having trouble. It is a pity, it seems like a race to see who has the fewest problems.
"Milan, in Italy, can do more, while outside of Europe they are not competitive."
Milan face Rafael Benitez's Napoli at San Siro on Sunday.
‘After Manchester City’s recent form, maybe they’re the underdogs against Manchester United!’ Former Red Devils defender on this weekend’s derby
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
