Levi Bellfield is currently serving two whole life sentences for murdering Marsha McDonnell, Amelie Delagrange and Milly Dowler.
He has applied for permission to marry a woman at HMP Frankland in County Durham, where he is incarcerated.
His fiancee has been described as a ‘besotted’ blonde-haired woman in her 40s.
A Ministry of Justice spokesman said: ‘An application has been received and is being considered in the usual way.’
However, it is understood that Bellfield’s application is still being processed and is not at any further advanced stage.
Former Justice Secretary Robert Buckland told The Sun newspaper, which first reported the news, that the proposed prison nuptials ‘beggars belief’.
He added: ‘Milly never got to see her wedding day. It cannot be right that he gets to have his.’
Prisons minister Victoria Atkins added: ‘This man is an evil, evil monster who has left a trail of despair and grief in his wake.’
She has ordered an immediate review into the request branding the proposed nuptials ‘absolutely appalling’.
But she did not say whether she could actually stop the wedding going ahead, citing Article 12 of the Human Rights Act and the right to marry.
The Sun said he is engaged to a female visitor and got down on one knee to propose in front of prison staff at HMP Frankland.
The 53-year-old killer would need the permission of the prison governor to marry at the category A men’s prison.
Bellfield was given a whole life term for murdering Ms McDonnell, 19, in 2003, and murdering Ms Delagrange, 22, and attempting to murder Kate Sheedy, 18, in 2004.
He was already serving this sentence when he went on trial for killing schoolgirl Milly, who was snatched from the street walking from school to her home in Walton-on-Thames, Surrey, in March 2002.
Bellfield was found guilty of abducting and killing the 13-year-old following a trial at the Old Bailey in 2011.
Earlier this year he also confessed to the murders of Lin and Megan Russell in Kent.
Bellfield worked as a car clamper in Chichester and across the south.