Senior managers at G4S only realised "eight or nine days ago" that they could not provide enough security guards for the London Olympics, the company's chief executive has said.
Nick Buckles told the BBC that problems in the recruitment and deployment process were only recently identified.
G4S will lose between £30m-£50m on the contract, which is worth about £280m.
Shadow minister Dame Tessa Jowell said the "integrity and resilience" of the security plan must be established.
Mr Buckles, who reportedly earns £1.2m per year in salary and bonuses, said the company accepted it had "underestimated the task of supplying staff to the Olympics".
"We deeply regret that and we are deeply disappointed. It was a daunting task to supply that number of staff in a short timescale.
"I began to know it was going wrong eight or nine days ago. Basically we are recruiting a large number of people and they are all working through a process of interview, two or three different degrees of training, licensing and accreditation.
"It is only when you get closer to the Games, you realise that the number is not as high as you expect," Mr Buckles added.
G4S signed the contract with Locog in 2010 to supply 2,000 security staff to work at Olympic venues.
In December 2011, the Games organisers asked the company to provide some 8,000 more, bringing the total to approximately 10,000 staff.
The value of the contract is about £284m. The company has predicted costs will reach approximately £330m, and estimated loses in the region of £30m-£50m.
Allowing for attrition - reduction in numbers due to factors such as sickness and no-shows - the company had to recruit nearly 14,000 people in total.
About 110,000 people applied for security jobs, and 50,000 were interviewed.
Mr Buckles confirmed on Saturday that 4,000 staff were "ready to go", having been trained and vetted, and received uniforms and instructions about their job.
A further 9,000 applicants were "in the process of being scheduled", meaning they had been trained and have security clearance, but did not have deployment information.
Mr Buckles described the recruitment process as "very complex", with applicants having to be "screened, SIA (Security Industry Authority) trained and licensed, role specific trained and accredited by the Home Office".
He would not confirm exactly what penalty the government was imposing on the company for failing to fulfil the contract. But he estimated the figure would be about £10m-£20m.
'Integrity and resilence'
The Commons Public Accounts Committee has summoned G4S, two government departments and 2012 organiser Locog to answer questions in September.
Mr Buckles has been asked to appear before the Home Affairs Select Committee next week.
Dame Tessa Jowell, the shadow minister for the Olympics, said a review into security at the Games should wait until after the event has finished.
She said the breakdown in security planning had been identified and remedied "in a way that the public can have confidence in."
But she added: "The focus now has got to be on nailing this down and ensuring the integrity and resilience of the security plan."
Elsewhere Clint Elliott, chief executive of the National Association of Retired Police Officers, said reports G4S asked his organisation to assist were wrong.
He said: "It is astonishing that G4S can't have recruited these people in a time of high unemployment and at a time when many police officers are being forced to retire early and are actively looking for work.
"G4S claim they contacted us but they never did. What G4S tend to do is rely on the old boys network and word of mouth to recruit people from our organisation - which is okay for small numbers but for 10,000 people you really need something a bit more substantial."
A spokeswoman for G4S confirmed it had not contacted the NARPO. Instead she said it had been directly contacting retired officers whose details were already on its own database.
Meanwhile, G4S has confirmed all security personnel working at the Games are fluent English speakers.
Earlier on Saturday, Mr Buckles said he was "pretty sure" the staff could speak English well "but I cannot say categorically as I sit here today".
The company has agreed to pay for the deployment of 3,500 extra military personnel, to plug the gap in security staff.
Mr Buckles added: "We are very, very grateful to the military for providing this support. To the individuals we are grateful that they are giving up time with family to come and help us.
"Together we will provide a safe and secure Games."