Former school caretaker Huntley, now 43, received two life terms for the murders and was sentenced to 40 years behind bars in 2005.
He has apparently said he is a ‘reformed’ character and believes he should be able to make a bid for freedom after 18 years of his sentence, which will be in just three years’ time.
Huntley, a former school caretaker, was first sent to HMP Belmarsh after being jailed for the 2002 murders of Holly and Jessica in Soham, Cambridgeshire.
He was later moved to HMP Wakefield and transferred to Frankland in 2008.
A prison insider told the Daily Star on Sunday: ‘Huntley sees himself as a reformed character and has been complaining about his 40-year sentence.
‘He has been banging on about it to fellow inmates and some of the prison guards.
‘Huntley has been telling people he is a different person and believes he should be allowed to apply for parole after serving 18 years – which isn’t even half his sentence.
‘He also somehow thinks that he should now be moved to a lower category prison, where the regime is more liberal.’
The insider added that Huntley had gone to a number of self-help classes and meetings at HMP Frankland to convince supervisors he was reformed.
But they said Huntley’s only concern is ‘for himself’ and that he has not showed any regret.
In HMP Frankland, Huntley has supposedly become friends with Yorkshire Ripper Peter Sutcliffe after bonding over their love of Planet Earth II.
The high-security jail also houses Milly Dowler’s killer Levi Bellfield and Mark Bridger, who murdered five-year-old April Jones.
Huntley has apparently never shown any remorse for killing Holly and Jessica and denied murdering them during his trial, claiming both deaths were an accident.
It was reported recently that Huntley had asked fellow prisoners to refer to him by a female name inside the all-male prison.
Huntley supposedly signed off a newspaper cutting for a friend with the name ‘Lian’.
The revelation comes after it was claimed last month the 43-year-old wanted a sex change so he could spend his life term in a female prison.
Holly and Jessica had been at a family barbecue in August 2002, when they went to walk to a nearby shop for sweets.
Their bodies were found nearly a fortnight later, around 14 miles away in a ditch near Lakenheath, Suffolk.
Huntley was sentenced to two life terms, with a minimum 40-year tariff.
MailOnline have contacted the Ministry of Justice for a response.