There a couple of way how you can see the existing claims of your ally.
You can see the claims of your ally on the map. Switch to "Diplomatic Map Mode" and then click on you ally, so you see his view of the world. His claims will show up as cyan circles around the systems (usually on his border).
This is properly the most efficient way since you can quickly get an overview on which systems your ally has claimed.
In the claim window hover with your mouse of the systems you suspect your ally has claims. You can see in the tooltip if the system is claimed by anyone.
Finally you can also see existing claims in the system window. Just hover with your mouse over the "Add claim" symbol and you can see any existing claims of that system in the tooltip.