The church says the attack took place on Thursday.
Security officials say the attacker struck the priest's head with a cleaver and fled the scene, but was later arrested. The officials spoke on condition of anonymity because they were not authorised to brief the media.
The motive was not immediately known.
Attacks on Egypt's Christians, who account for about 10% of the country's 94 million people, have surged in recent months, with a series of suicide bombings claimed by the Islamic State group killing more than 100 since December.