The arson and graffiti are suspected to be a so-called "price tag" attack in response to the government's recent evacuation of a Jewish settlement in the West Bank, Haaretz reported.
Monks residing at the monastery noticed the burning door and extinguished the flames before notifying police.
One monk told Haaretz the attack was the first act of vandalism in the monastery's 122-year history.
"I know there is some tension, but I don't understand why it has to do with us," the monastery's abbot said.
Jerusalem's acting district police commander said a special investigation team would look into the incident, Ynetnews.com reported.
Police have been preparing for "price tag" actions by right-wing activists following the evacuation of the West Bank outpost of Migron.