Lee Cyrus absconded from HMP North Sea Camp in Boston, Lincolnshire, on October 9, and has been on the run throughout Scotland and England ever since.
The 47-year-old, who is believed to be using public transport to travel around, is known to have been in the Tayside area and Preston over the last few weeks.
Detectives investigating rapes in Perth on October 15 and Aberdeen on October 27 have refused to rule him out of their inquiries and Tayside Police say they have received reports of more than 100 sightings of the fugitive, who has previous convictions for burglary, assault and sex offences against a young girl.
However, Tayside Assistant Chief Constable Angela Wilson has now admitted officers "really don't know" where Cyrus is.
She said: "We know he has definitely been in Tayside and there was a confirmed sighting in Preston, but none of the other reported sightings have been confirmed. He could be anywhere between Scotland and England."
Wanted posters have been distributed to police forces throughout the two countries and will appear in shops, community centres and bus and train stations.
Earlier in the week, there were also reported sightings in Aberdeen and Edinburgh, but these both proved to be false.
The unconfirmed sighting in Aberdeen, in the tiny coastal town of Boddam, sparked panic in the community as dozens of police officers and sniffer dogs descended on the quiet village.
The search was sparked by two patrolling police officers who noticed a cyclist matching Cyrus's description. However, it was stood down after officers located the look-alike and confirmed it was not the absconder.
Despite the sighting turning out to be false, detectives investigating a rape in the Powis area of Aberdeen on Saturday morning have refused to rule Cyrus out of their inquiries.
The 22-year-old woman was attacked as she walked alone in a shopping area.
Tayside Police are also not eliminating him from a probe into the rape of a woman in her home in the Potterhill area of Perth where the 23-year-old was attacked as she slept.
A confirmed sighting of Cyrus was reported on CCTV in his hometown of Preston two days later and a man matching his description is also believed to have made sexual advances towards two 15-year-old schoolgirls in the town.
The last confirmed sighting was also in Preston on October 19.
Ms Wilson said Tayside Police has a "significant number" of officers involved in the search for Cyrus, who is also known as Ivan Leach.
She added: "He's obviously lying low and trying to avoid capture, but eventually he'll have to go to a shop and buy food or cigarettes or something. You can't lie low for long periods of time without making use of those services.
"He is a dangerous individual and the public should rightly be concerned he is at large. We would ask them to be alert, aware, vigilant and security conscious.
"If you seem him contact the police immediately."
The police chief, who said Cyrus has no known links to the Tayside area, added that detectives are pursuing a number of lines of inquiry in relation to the rape investigation.
Cyrus, who was serving a life sentence for robbery when he absconded, is described as being about 5ft 11in tall, of stocky build, with cropped or shaved hair and hazel eyes.
Crimestoppers is offering up to £3000 for information leading to his arrest and conviction.
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