The Sun and the Daily Mail have jointly written to the Parole Board and the Justice Secretary.
The letter says that unless the reasons behind the decision to recommend the rapist's release from prison are published within seven days, lawyers will apply for a full judicial review to have that decision overturned.
It says the report is a "matter of profound public interest" and keeping it secret is a breach of open justice.
The controversial Parole Board decision to approve his release later this month, following his minimum eight-year term, has led to a moral outcry - with lawyers for his victims describing the move as "extremely distressing".
Black cab driver Worboys was jailed indefinitely in 2009 for drugging and sexually assaulting women passengers.
He was convicted of 19 offences relating to 12 victims but is suspected to have been linked to 102 complaints in total.
It is understood that a fresh allegation of historical sexual assault has been made against the 60-year-old.
The incident, which was reported to the police earlier this month, is reported to have taken place in 1997.
Scotland Yard is now investigating the claim against Worboys, who has not been arrested.
Over the weekend, he was reportedly transferred from HMP Wakefield in West Yorkshire to HMP Belmarsh in southeast London.
Justice Secretary David Gauke is facing demands for Worboys to be kept out of the capital - the area in which he preyed on his victims.
Mr Gauke said he had stressed to officials "the need to ensure that the concerns of victims are at the heart of the process" in deciding Worboys' licence conditions.
A source told Sky News it would be "completely untrue" to suggest Worboys will be released imminently.
Upon release, Worboys would be expected to spend several months in a probation hostel, which would be staffed 24 hours a day and impose a night-time curfew.
Last week, Mr Gauke revealed the Government will not be seeking a judicial review of the decision to release Worboys.
Lawyers for two victims have said they intend to pursue the case in the courts.