According to the Corriere della Sera newspaper, the Vatican argued in a formal note that the bill violates the Concordat, the treaty between Italy and the Holy See.
'It is an unprecedented act in the history of relations between the two states - or at least, there are no public precedents,' the daily said.
Pope Francis' de facto foreign minister, Archbishop Paul Richard Gallagher, delivered a letter, or 'note verbale', to the Italian embassy to the Holy See on June 17.
The letter suggested that the Zan law would breach the Concordat by curtailing Catholic freedom of belief and expression.
This is because Catholic schools would not be exempted from an obligation to take part in a newly-created national day against homophobia, lesbophobia and transphobia.
The letter also expressed concern that Catholics could in the future face legal action for expressing opinions against LGBTI rights, Corriere said.