This question came up in the Online 10-day Retreat Nov 3, 2020, guided by Delson Armstrong. Day 6 was part of a daily 30-minute discussion on the suttas, the Dhamma talk and reflections.
At that time there is no identification with that intention. So, in the case of an arahant -somebody who is fully realized and fully enlightened -, ignorance is no longer conditioning or hindering the Formations. And craving and conceit are no longer fetters within the Formations. Ignorance is replaced by Right View. Whenever the Formations arise in the arahant, it is due to contact, kammic contact. So, they are experiencing the effects of old Kamma [created] prior to full awakening.  The Formations, as kammic carriers, will take these kammic seeds or the kammic effects of the old actions, old thoughts, and old deeds and speech from prior to full awakening, and they will carry forward into the consciousness, which will drive forward in intention. But all throughout that process, there is no sense of involvement of a of a personal self. It is just being seen with mindfulness as a series of impersonal processes.
When someone like the Buddha or an arahant monk in the suttas says: “It occurred to me”, they are using conventional language that the thought came into mind. But the thought came into mind because of the Formations, and then the intention to say for instance maybe I'll go out for alms today at this place. Or, maybe I'll go visit these people at this place.
Even though it is using language that is very conventional. the understanding behind it, deep down, is that it's all impersonal. There is no taking personal the intention, and since there is no craving or identifying with the intention, the intention is completely void of any form of fettered Formations, void of any craving, void of any identification.
So, when that intention drives forward, it will continue on without creating any potential for craving to arise, for any potential for identifying with the feeling that arises at the level of thought, or at the level of the sensory experiences.