I know that some banks have an email address right on their website to send the message to. I believe they're usually under the 'contact us' sections, but the ones I've seen are specifically for fraud situations.
I've done it before (I've gotten several phishing emails myself), and have always gotten a same-day response confirming the receipt and then the a reiteration of the bank's policy if some information was compromised.
Otherwise, you could probably contact their 24 hr reps, if the bank is big enough, and get to the internet people to ask for the email address to forward it...
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