Embattled security group G4S is bidding for contracts worth more than £2bn to run five prisons. The revelation is likely to trigger a furore among the company's increasingly long list of critics and renew the often feverish debate about the extent to which the private sector should run public services.

News that the army and police have had to be drafted in to provide security at the Olympics because G4S is unable to fulfil its contract to supply 10,000 stewards has triggered calls for the company's chief executive, Nick Buckles, to resign.

Last week, MPs on the public accounts committee expressed concerns that management costs associated with the contract had soared from £10m to £125m, prompting concerns that the taxpayer was not getting value for money.

Now unions are questioning whether G4S's failures over the Olympics should preclude it from running more prisons.

G4S already operates Parc, Altcourse, Oakwood, Birmingham and Wolds prisons. But, under the government's Competition Strategy for Offender Services, announced in July 2011, it has emerged that it is also bidding to run Northumberland, Hatfield, Lindholme, Moorland and Onley prisons as well as to renew the contract for Wolds.

The probation officers' union, Napo, estimates each contract could see G4S paid as much as £40,000 per prisoner a year over the life of the 15-year contracts, which will be awarded in November. If G4S were to be successful in all of its bids, it is estimated that the total value of the contracts would exceed £2bn.

Supporters of the private sector say bringing in outside contractors will save the taxpayer money by helping to cut out inefficiencies associated with the public sector. But critics question whether G4S will offer value for money, given recent events. They also claim there is little oversight of how public sector contracts are awarded to private companies – or redress for taxpayers if costs soar.

"The fiasco involving G4S and the Olympics has shown again how there is little if any accountability to the state and how contracts are far from transparent," said Harry Fletcher, assistant general secretary of Napo, who claimed an absence of a "level playing field between the private and public sector".

The Prison Officers Association has written to justice secretary Kenneth Clarke saying the Games fiasco highlights the perils in "relying solely on private firms for security". Steve Gillan, the association's general secretary, told Clarke: "I urge you strongly, to think long and hard before any more prisons contracts are handed over to private firms to engage. I fear this is a race to the bottom – and prisons are safer and more accountable when they are run by the public sector."

Questions are also being asked about the role played by Serco, the private contractor drafted in to protect Britain's ports during the Olympics. In mid-May, concerns about queues at ports forced the Home Office to bring in private guards to operate radiation screens designed to detect "dirty bombs" being smuggled through major ports.

The scheme is part of Programme Cyclamen, a joint initiative between the Border Agency and the Home Office. Under Cyclamen, most sea and rail traffic , including containers, freight, vehicles and passengers, is screened as it goes through UK ports.

A memo from a senior Border Force official reveals that some guards employed by Serco at Dover cannot be contacted because of problems with their radios. It also observes that vehicle screening problems that began before Serco became involved have continued. It notes that, because of "communication issues that need to be resolved", projected cost savings arising from the hiring of Serco staff are not being realised and that "there have been occasions when BF [Border Force] officers have had to correct errors by Serco staff when capturing alarmed vehicles".

Serco will come under further scrutiny on Monday when the home affairs select committee condemns the backlog of outstanding cases that the agency is failing to clear. A Border Force spokesman said: "Security of the UK border remains our top priority and all staff are fully trained to operate their role. Serco staff work alongside Border Force officers and will be expected to adhere to the same high standards as our own staff."