Be careful when you say Eurodollar! Eurodollar is a US dollar deposit outside of the US, so it's a short-term interest rate product not a FX product.
Euro FX, which also trades on the CME, is the futures contract on the EUR/USD currency pair.
EUR/USD pair is the spot market while the Euro FX futures is the forward market. So while the two do move together the futures have a carry component and thus basis risk.
