The decision by the Anti-Violence and Racism in Sport Commission comes in light of comments by the Spanish sports minister Miguel Cardenal, who is looking for a stiffer sanction. The case will now be judged by an appeals court.
Brazilian Alves received widespread support across the soccer world after picking up the banana and biting into the fruit as he prepared to take a corner during the April 27 La Liga match.
The defender later told Brazil's Radio Globo that Spain was "very backward" in its approach to racism.
"The decision has been made to appeal the 12,000 euro fine to the Administrative Tribunal for Sport," read a statement on Thursday from the Anti-Violence and Racism in Sport Commission.
"The commission does not agree with the contents of the verdict" and that "it should be revised given the severity of what took place".
The fan, David Campayo, was given a 6,000 euro fine and a two-year ban from attending sports events. Villarreal already gave him a lifetime ban immediately after the incident.
Earlier on Thursday, Cardenal told TVE television: "We think it is a situation that demands a stronger penalty. We need to send a stronger message to the public."
Cardenal said any form of racism was "abhorrent" and stern action always needed to be taken.
"We must always have zero tolerance and fight against racism with all our powers," he added.
Spanish football is trying to improve its image but was not helped by allegations last weekend from Levante's Senegal midfielder Papakouly Diop, who complained of racist abuse by visiting Atletico Madrid supporters.
- Sports & Recreation
- La Liga