As you can see in the screenshot above the eyeball cannot be seen through the glass, but the iris can.
I don’t know why the eyeball cannot be seen, but I have altered the properties of the glass through some testing and realised a solution to this problem, to add a BSDF Transparent node to the Node Editor and place it before the last node before the output. If I placed it as the final node to the output it will go completely transparent, which might be useful, but I would just remove the object in that case.
Here is the screenshot of it resolved:
As you can see, it looks better now you can see the whole eyeball.