These are fungible contracts. Sellers get first notice. Sellers get to choose the deliverable from the exchange basket to include the exchange conversion factor. Buy a cash bond in the cash market, sell it to the exchange.
Why this rant?? CBOT volume flips about the first trading day in the delivery month, and EUREX about the fourth day. This is usually a gradual transition lasting about two days, as these are fungible contracts. Eurodollars are indexed, so you can hold them for the final settlement value.
Keep your eye on the open interest and the cheapest-to-deliver issue.