When did Gandalf Choose Bilbo?

[Morsul the Dark]

This question is the end result of a long train of thought. I was looking into the intervening years between the fall of Erebor and the quest. I know they ended up in the Blue Mountains for a while but Thorin, in particular, never gave up the thought of taking it back.

So a bit of Yada Yada Gandalf gets the key and map decides on burglary and just seemingly decides on a Hobbit and Bilbo in particular. The thing is he chooses Bilbo(seemingly) in the morning and that evening the dwarves arrive. Gandalf must’ve already settled on Bilbo, or the very least a Hobbit(and Bilbo specifically seeing as the dwarves all go directly to BagEnd) and had the dwarves already on the way. I’m reminded of Gandalf keeping Beorn busy with a story slowly marching the dwarves up two by two, they arrive to Bilbo’s in a similar fashion just a couple at a time and he lets them in he probably would have more forcefully reject 13 at once, it seems a very Gandalf arrangement.

So the question after that ramble when did Gandalf settle on Bilbo?