What we've seen in the anime is Goku first got god ki when he became super saiyan god (red) after making the ritual where 5 saiyans of pure heart surrounded him.
Vegeta first got god ki only (not in super saiyan) when he was sent to train to another dimension by Wiss along with Goku
there Vegeta tells what they call god ki is ki which doesnt overflow from his body
In this other video, after being told by Wiss not to overflow their ki, both Goku and Vegeta get a taste of super saiyan blue, which supposedly is "having the power of a super saiyan god (god ki) and then becoming a super saiyan"
So, a requisite to have god ki is to learn how to control ki for it not to overflow from the user's body. Personally, I also believe you need a certain minimum power level to attain god ki, otherwise why Wiss made Vegeta train and raise his power level before telling him how to use and get god ki (Additional note: Wiss also mentions "he is reaching the level where he can sense god ki").
Unless you are a special case, such as Kamisama (Dende) or Supreme Kai, who can feel other's god ki and therefore have god ki, because they were assigned gods or they were born gods, when they have a low or not so great power level (it's stated than only gods can feel god ki)
And for the manga, I dont remember anything relevant about god ki. Usually Toyotaro in the manga tries to fix the mess done by the anime or movies regarding different contradictory concepts with some additional explanations , but this wasnt the case. (sometimes he also makes more mess than coherence)