A word used to describe a fish trying to eat bait bit by bit

To turn several good comments into an answer: nibble.

[Merriam-Webster]
1 a : to bite gently
1 b : to eat or chew in small bits

I'm thinking back to fishing trips with my grandpa. When he described this to us, and when we would talk about it with him or with each other, there are a few ways of expressing this concept that I can remember.

1) "He's nibbling at it."

OR

2) "He's picking at it."

OR

3) "He's trying to steal your worm."

I agree with the other comments and the answer by @Roger - "nibble" is probably the most-commonly-used word, but I wanted to add these alternatives. The second and third options - especially the third - seem to express all the nuances of

tr[ying] to eat the bait bit by bit

a bit better than simply using, "He's nibbling at your worm." That's my feeling, anyway. Often, the comment with nibbling expressed an expectation that the fish would soon take a big bite and get caught. The comment with stealing expressed the idea that the fish knew that the worm was on a hook and was therefore trying to get all the meal he could without touching the hook. The comment with picking was somewhere in between.

By the way, it seemed that we always said something like, "He'll get greedy, eventually," after the preceding statements.

P.S. I guess all fishes were male until we gutted them and found out differently.