One of the most famous Islamic monuments in Europe, the grand stone edifice dominates the old centre of the city in the southern Andalucia region.
It is listed by UNESCO as a world heritage site, a jewel of Islamic architecture with later Christian additions, located not far from the old Jewish quarter.
Locals call it the "Mosque-Cathedral" -- but critics accuse the Catholic Church which runs the site of covering up the Islamic part of the building's past.
Visitors to the site are issued with tickets and information leaflets that call it a cathedral, with just minor passing references to the mosque.
"The Mosque-Cathedral is the cultural heart of the city and its principal tourist business," said Miguel Santiago, a spokesman for the campaign group Mosque-Cathedral, Heritage of All.
"Changing its name goes against the tourism interests of Cordoba and, worst of all, against its historical, artistic and symbolic essence."
A spokesman for the local cathedral council, Jose Juan Jimenez, said the Church has been the "legitimate owner" since 1236 when Christian King Fernando III had it consecrated.
A mosque had been built there on the site of an earlier Visigoth church from the eighth century when Cordoba was held by Islamic Umayyad rulers.
Fernando converted it into a cathedral when he reconquered the city in the 13th century.
In 1998, a change in the law by the conservative Popular Party government permitted the church to register certain places of worship as its own property even without possessing title deeds.
In 2006, the Cordoba bishopric registered the cathedral as its own property for a sum of 30 euros.
"Until 1998, all the bishops of Cordoba had considered themselves users and protectors of the monument, but never its owners," said a spokesman for the group, Miguel Santiago.
"When they start to consider themselves its owners, they start to change its history, even eliminating its name."
The campaigners say the act of consecration did not grant the church legal rights to the property.
They say leaflets that downplay the building's Muslim past and that talk of a mere "Islamic intervention" or "Islamic traces" do a disservice to the city.
Local Muslims have kept a low profile in the dispute.
But Kamel Mejelef, president of the Cordoba Muslims Association, admitted: "It hurts us when they try to distort the history of the building, erase the name of the mosque.
"When people visit Cordoba, they come to see the mosque, not the cathedral."
The campaign group is demanding the building be run jointly by the church and local authorities as a conservation site.
"Priests know all about mass, but I doubt they know much about tourism and culture," said Santiago.
The Church insists it is able to preserve the Mosque and its history, saying it has done so for the past eight centuries.
"This place of worship is a cathedral, which was formerly a mosque and before that was a Visigoth basilica," said Jimenez.
"We did not do what the Muslims did with the basilica, which was destroyed at the time. We respect, care and conserve. Anyone who visits can see the Islamic traces."
Inside, Moorish columns and alcoves stand alongside the soaring Cathedral nave that was added in the 16th century.
An online petition demanding that Spanish authorities and UNESCO intervene has gathered 387,000 signatures on the website change.org.
The city hall is worried about the effect of the row on Cordoba's image.
"Before this goes any further, we want to clear up the matter and give an image of Cordoba as an open city, a city of consensus in which understanding and coexistence prevail," said its mayor, Jose Antonio Nieto.
Nevertheless, last year as the row rumbled on the site received more than 1.5 million visitors, nine percent more than in 2013. The bishopric took 10 million euros in entry fees.
"As a cathedral, it is like all the others in Andalucia," said Maria Angeles Querol, a specialist in heritage management at Madrid's Complutense University. "As a mosque, it is unique."