Eurodollar bank does not usually have an account with the FED. Otherwise they are part of normal dollar system.
Fun thing is - swift does not actually move the money, it's a messaging protocol to agree on how the movement is done.
Discussion
well, not directly, but there must be a corresponding bank (account) which does. Otherwise they couldn't endup in a US bank, could they?
(I know that swift is more of a protocol).
How they settle balance sheets is probably up to them, isn't it? It's really been a while since I read how all that works.
Yes. Banks have accounts with other banks with balance. They can top them up short term with repo, long term with fx trade. It's basically a deposit account called nostro account. The key here is that eurodollars switch to dollars backed by reserves (central bank money) there.