Private security firm G4S told the Home Office it would be able to resolve its problems supplying Olympic guards last month, Theresa May said on Friday.
On a tour of the Olympic Park's security centre, the home secretary said G4S, the world's second largest private sector employer, told officials that any problems were temporary and would be resolved.
"In early July it was clear that there were some initial problems which G4S said they would resolve," Mrs May said.
"And crucially it was not until 11 July that G4S finally said, 'Actually we can't resolve those initial problems, we won't be able to provide the personnel'."
She went on: "I haven't been at all selective. What I've been is absolutely clear with the House of Commons and others about how these things developed."
Asked what the Home Office was told at a meeting with G4S and Olympic organisers Locog on 27 June, Mrs May said: "What happened was there were some early signs of a problem with rostering staff for G4S.
"It was clear that G4S felt they were capable of dealing with that, that it would be resolved, and it was on 11 July as the chief executive of G4S told Parliament, the home affairs select committee, that G4S said, 'Actually, we now believe we cannot produce the staff we were contracted to produce'."
Mrs May went on: "What I've seen here is a very organised control room, people absolutely focused on their role of providing the safety and security for the Olympic Games.
"And, crucially, there has been no compromise in security given what has taken place in recent days and I can see that, there's a visible example of that, here in this control room.
"We are constantly monitoring, constantly looking at how they are ensuring that safety and security for the Games so the people who come to these Olympic Games enjoy them for the great sporting event that they are and be able to feel safe and secure."
With a further 1,200 troops put on 48-hour stand-by yesterday as possible support for the 17,000 servicemen and women already involved in Games security, Mrs May added: "The government needs to be sure we can be ready for any contingencies."
Asked if G4S should be barred from taking on any government contracts in the future, she said the government was working with the firm on both the Olympics and Paralympics, as well as a number of other contracts.
"G4S will still be providing a significant number of venue security personnel," Mrs May said.
During the visit to the control room with Scotland Yard Commissioner Bernard Hogan-Howe and London 2012 chairman Lord Coe, Mrs May saw G4S staff and others at work in front of a bank of more than 30 large screens covering the park.
She also met Colonel Gary Wilkinson, the venue's senior military representative, and Superintendent Neil Seabridge, the Met's bronze commander at the park.
Mrs May said the issues with G4S supplying enough guards initially looked like "teething problems".
"They were consistently telling us, up until 11 July, that they could produce the number of people required for the Olympic Games," she said.
Asked if G4S should have acted sooner, Mrs May said they believed they had identified a "temporary problem which was capable of resolution".
Olympics supremo Lord Coe insisted there were no concerns over security.
On Thursday, Mrs May was accused of giving MPs a "selective account" about when she knew G4S were having problems supplying enough guards.
The home secretary admitted the government and organisers Locog knew there were problems as early as 27 June.
It came after the company's under-pressure chief executive Nick Buckles told MPs he was told about the problems a week later on 3 July and Mrs May told the Commons the "absolute gap in numbers" was not known until 11 July.
A further 1,200 troops have been put on standby to provide Olympics security as the fallout from the shambles continued.
But the government said the numbers of staff to be provided by the company were rising, and there was currently no need to deploy more military personnel.
Shadow home secretary Yvette Cooper said Mrs May "needs to explain urgently how she justifies having given Parliament and the public such a selective account, and why the Home Office were so slow to respond".
The details were revealed by Mrs May in a letter to the Commons Home Affairs Select Committee.
Labour MP Keith Vaz, the committee's chairman, said: "The home secretary told the House that she only became aware of a shortfall on 11 July.
"However this letter clearly states they were warned of a possible shortfall in guards on 27 June at the Olympic Security Board, two weeks before."
He called for any updates on the figures contained in monthly internal assurance reports to be released and for an explanation of "why they did not ring alarm bells sooner".
While 750 troops were put on 24 hours notice last month, the extra 3,500 servicemen and women who will plug the gap left by G4S were not called in until two weeks later after the company confirmed its "absolute" shortfall.
The number of military personnel involved in Olympics security is now 17,000, including 11,000 who will help secure Games venues, with the rest working in specialist roles.
A further 1,200 are on 48 hours notice to move, in case they are needed.
In a statement last night, a G4S spokesman said the firm "believes that, even if some or all of the additional troops now on standby were to be deployed, the overall losses to be incurred on this contract would remain within the previously stated estimate of £35-£50 million."