They said that senior Spanish Catholic clergy had earlier rejected requests for Muslims to be allowed to prostrate themselves inside the Cathedral.
Mansur Escudero, the board's secretary general, said security guards often stop Muslim worshipers from praying at the old mosque.
He said: "There are reactionary elements within the Catholic Church, and when they hear about the construction of a mosque, or Muslim teachings in state schools, or about veils, they see it as a sign we are growing and they oppose it."
Mansur said Muslims came from around the world to see Cordoba's Cathedral, which is still commonly known as the Cathedral-Mosque.
The Roman Catholic cathedral had originally been a mosque but was converted into a cathedral in the 13th century.
The mosque itself was built on the site of the earlier cathedral of St Vincent which was demolished by Cordoba's Muslim rulers following the Islamic invasion and occupation of parts of southern Spain in the eight century.
In December, Spain's Catholic Bishops Conference released a statement, quoted by newspaper ABC, saying it "did not recommend" Muslims prayed at the Cathedral and was not prepared to negotiate the building's shared use with other faiths.
Spain's last Muslim territory fell with the conquest of Granada in 1492 after almost eight centuries of Muslim rule.
Today, more than a million of Spain's 44 million people are Muslims, many of them recently arrived immigrants from North Africa.