Levi Bellfield has confessed once again to killing Lin Russell and her daughter Megan in 1996.
Gwynedd artist, Josie Russell, was nine when she and her mother, Lin, and sister, Megan, six, were attacked with a claw hammer.
The family had recently moved from north Wales to Kent when the attack happened. Josie and her father, Dr Shaun Russell, subsequently moved back to the area.
Bellfield said last year that he attacked Josie and killed her mother and sister, but retracted his confession. Now his lawyer says he has once again admitted murdering Lin and Megan.
Michael Stone was jailed for life for the murders in 2001, but now Bellfield's lawyer says her client has confessed, because he "needs to take responsibility".
Paul Bacon, Stone's solicitor, says police need to investigate the case again.
Bellfield is serving a whole life sentence for the murder of Milly Dowler, 13.
He has also been convicted of the murders of Marsha McDonnell and Amelie Delagrange, and the attempted murder of Kate Sheedy. He will never be considered for parole.