Your dirt is too far from the grass.
According to the Wiki grass can only spread up one level at a time, though it can spread down three levels, and, if I'm reading it correctly, only one block sideways.
So replace that cobblestone in between with dirt, you can switch it back once the grass has spread to the top.
I believe the grass will spread faster if it has a wider path.
If you have any tool with Silk Touch you can move some grass blocks and place them closer to where you want the grass to spread.
Neither the biome nor if there is water nearby should matter.