Former detective superintendent Steve Fulcher caught the 52-year-old following the abduction of Sian O'Callaghan from outside a nightclub in Swindon in March 2011.
The taxi driver would go on to confess to murdering both Miss O'Callaghan and another woman, Becky Godden, and now Mr Fulcher believes Halliwell's links to the area where Miss Lawrence vanished should be investigated.
Miss Lawrence was last seen on March 18 2009 and was reported missing the following day after she failed to turn up for a shift at work at the University of York.
Her father Peter is a Christian who's campaigned on behalf of missing people, for example to have the law changed so relatives can manage their accounts in their name so debts don't build up.
The Archbishop of York Dr John Sentamu marked 2000 days of Claudia's disappearance in 2014.
Martin Dales, friend and spokesman of Peter, said: "I think the retired officer in question has a detailed knowledge of things. The overriding point is really whoever has done whatever to Claudia, it has been an incredibly long time.
"It's a heartbreaking period to go without somebody and we hope this will come to some conclusion - whether this is it is for other people to find out.
"It's got to be thoroughly checked out."
