In a 2007 interview JK Rowling once said:
Also, James and Lily turned him down, that was established in Philosopher's Stone. He wanted them, and they wouldn't come over, so that's one strike against them before they were even out of their teens.
But I find this odd. Why would Voldemort wish to recruit a Muggle-born witch to the Death Eaters?
'The Death Eaters can't all be pure-blood, there aren't enough pure-blood wizards left,' said Hermione stubbornly. 'I expect most of them are half-bloods pretending to be pure. It's only Muggle-borns they hate, they'd be quite happy to let you and Ron join up.'
'There is no way they'd let me be a Death Eater!' said Ron indignantly, a bit of sausage flying off the fork he was now brandishing at Hermione and hitting Ernie Macmillan on the head. 'My whole family are blood traitors! That's as bad as Muggle-borns to Death Eaters!'
'And they'd love to have me,' said Harry sarcastically. 'We'd be best pals if they didn't keep trying to do me in.'
Harry Potter and the Half-Blood Prince - p.227 - Bloomsbury - Chapter 12, Silver and Opals