Ivan Leach, also known as Lee Cyrus, 47, went missing from North Sea Camp open prison in Boston, Lincolnshire, after he failed to return from day release on October 9.
He is believed to have carried out a very serious sexual attack in the Tayside area since he absconded, although police said further details could not be released for legal reasons.
Police are also investigating reports that he may have made sexual remarks towards two young girls and encouraged them to participate in sexual activities with him in Preston 10 days ago.
Detective Chief Inspector Ian Dawson, of Preston Police, said: "Leach is an extremely dangerous and predatory individual who poses a serious threat to members of the public.
"He has previous convictions for burglary, sex offences against a young girl, assault and robberies in which he has targeted elderly people in their own homes.
"We believe he could be sleeping rough and travelling extensively throughout the country.
"I would remind people that he is violent and should not be approached."