A former senior executive at the National Australia Bank has been arrested and charged after travelling from Melbourne to Sydney to be interviewed by police over an alleged $40 million fraud engulfing the bank.

Rosemary Rogers, the former chief of staff to former NAB boss Andrew Thorburn, presented herself at Surry Hills police station about 11am on Tuesday, after flying of her own accord to Sydney.

Shortly before 3pm on Tuesday the 43-year-old was charged with 56 counts of being an agent corruptly receiving a benefit, and dishonestly obtaining financial advantage by deception.

Police allege Ms Rogers received bribes in the form of paid personal expenses, to the value of more than $5.4 million from a contractor to maintain the contract and approve inflated invoices.