Disgraced Edwards, 62 - the corporation's highest paid newsreader earned more than £200,000 from when he was arrested in November to when he left the BBC in April.
His pay during that time also included a £40,000 pay rise.
Last night the use of licence fee payers' money was slammed by campaigners who also called for BBC Director-General Mr Davie to resign.
Mr Davie said the BBC was unlikely to recoup the money it paid Edwards.
Asked about the newsreader's pension he said it was "very difficult to claw back, nigh on impossible".
He added: "These are unfortunately the specifics of how it works. When it comes to pay, again, legally challenging, but we'll look at all options."
But he defended the BBC's decision not to sack Edwards, despite knowing he had been arrested on "suspicion of serious offences".
Edwards pleaded guilty on Wednesday to three counts of making indecent images of children.
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