YMMV, however National does have it as stated policy that Debit cards (with major logos) are accepted upfront if and only if presented along with proof of a round-trip ticket. Failing that, it can also be accepted for payment upon return. To wit:
Debit Cards:
Debit/check cards are considered to be any non-credit card bearing the VISA, MasterCard or Discover Card logo. Any other non-credit card without the VISA, MasterCard or Discover Card logo is not accepted.
When renting in the US, a credit card in the name of the renter must be presented at the time of rental; otherwise, Debit/Check cards are accepted along with proof of a confirmed round trip travel ticket (airline, cruise ship or train). Without proof of roundtrip ticket, debit or check cards are only accepted as a form of payment at time of return.
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I wouldn't be surprised if you encounter resistance to doing it though.
I am 100% sure however that you cannot have the debit card on file for your EC account.