The soaring number of sex offenders is putting pressure on space available at Birmingham prison, a report has said.

Historic sex cases being solved, the rise of internet pornography and higher rape prosecutions are causing difficulties for the privately-run prison, its Independent Monitoring Board said.

In the first annual report since the prison was privatised, the board said it had written to Justice Secretary Chris Grayling asking him to investigate urgently. The board said: "Suitable numbers of prison places need to be provided at establishments which can offer appropriate rehabilitatory programmes for these offenders."

HMP Birmingham, which can cater for up to 1,450 adult male prisoners at a time, became the first UK jail to be taken over by a private company in October 2011 when G4S secured a 15-year contract.

The board added that there was a problem in transferring sex offenders to more appropriate sites where they can take part in courses which are not available in Birmingham.

The board said the prison's G wing now exclusively housed sex offenders and that bullying had reduced because of it. However, it said the high number of sex offenders at the prison meant some were being held elsewhere in the jail until space on G wing became available. The transition into the control of G4S had "not been easy", the Board added.

It also asked the director of the prison, Pete Small, to investigate problems with pests including rats, mice and cockroaches, most noticeably in the kitchens.

Mr Small said: "The board's report covers a 12-month period up to last summer, and since that time action has been taken to address many of the issues raised in the report.

"There are undoubtedly challenges to overcome at Birmingham. The length of the privatisation process was a difficult time for staff, and the readjustment to new management takes time."

A Prison Service spokeswoman said: "The report by the Independent Monitoring Board at HMP Birmingham will be fully considered by Ministers and we will respond in due course. We are increasing the availability of treatment programmes available across the estate for sex offenders."