The News of the World publisher has also offered a vast sum of money to charity in order to make amends for hacking the murdered schoolgirl's voicemail and giving the family false hope, BBC News reports.
Journalists tapped into Dowler's phone in 2002 shortly after she disappeared, deleting messages left on her mobile phone when her inbox filled up.
The News of the World was subsequently shut down by News International as revelations about the practice of phone hacking spread.
According to The Guardian, the Dowler family's lawyers were hoping for a settlement figure closer to £3.5m.
Both News International and the Dowlers' solicitor have declined to comment.
> Milly Dowler family shocked by News of the World hacking revelations