"Lay Spain", an atheist campaign group, claims these perks are worth up to 3 billion euros each year.
The opposition Socialists meanwhile claim that many church properties could be taxed that are classified as exempt at the moment.
The opposition Socialist Party has asked the 2,500 councils it controls to find ways to charge the Church property tax.
Under the law, only properties devoted to religious worship or social services are exempt from tax, but in practice other church-owned buildings - such as the palaces in which many bishops live - are also exempt.
Cardinal Antonio María Rouco, president of the Spanish Bishops' Conference, said the Church would abide by the law but cutting tax benefits might affect charitable work.