Relatives of a Brazilian man shot dead by police who mistook him for a suicide bomber were briefed by officers two days later, Scotland Yard says.
Mr Menezes' cousin, Alessandro Pereira, gave a letter to Mr Blair
The cousins of Jean Charles de Menezes were told he had not run into London's Stockwell Tube station and had not vaulted the ticket barrier, it said.
They were also told he was not wearing a padded jacket, contradicting reports.
Scotland Yard issued a statement after a "positive" meeting with senior Brazilian officials on Monday.
Wagner Goncalves, of Brazil's federal prosecutor's office, and Marcio Pereira Pinto Garcia, of its ministry of justice, flew to the UK on Monday to look into the circumstances surrounding the killing of the 27-year-old electrician.
They met members of the Metropolitan Police, including Metropolitan Police Commissioner Sir Ian Blair, in a discussion chaired by Deputy Assistant Commissioner John Yates.
The are also due to meet representatives from the Independent Police Complaints Commission (IPCC), which is investigating the case.
In a statement following the meeting, Scotland Yard said it had been "positive and constructive" and had covered the protocols of the investigation, but had not gone into specifics because of the IPCC inquiry.
It also said: "Mr Yates outlined the details given by police to the family and Brazilian Consulate officials in the aftermath of this tragedy.
"He said that on Sunday, July 24, Mr de Menezes' family in the UK, his cousins, were briefed by police that Mr de Menezes did not run into the Tube station, that he used a ticket to get through the Tube station barrier - specifically that he did not vault the barrier - and that he was not wearing a padded jacket or carrying a bag."
He also reiterated the apology made to the family for the death of Mr Menezes, Scotland Yard said.
Mr Menezes' mother has said action must be taken against the Metropolitan Police.
"They took my son's life. I am suffering because of that," she told BBC News.
"I want the policeman who did that punished. They ended not only my son's life, but mine as well."
On Monday up to 200 supporters of Mr Menezes' family attended a protest vigil at Downing Street to mark a month since he was shot.
His cousin handed in a letter to the prime minister calling for a public inquiry.
Leaked documents from the IPCC investigation into the death last week appeared to contradict initial police and witness statements about the incident.
Scotland Yard had been quoted as saying that Mr Menezes' "clothing and behaviour" added to suspicions on 22 July - a day after four failed bombings on the transport network - that he was a suicide bomber.
But a leaked photograph of the body of Mr Menezes on the Tube showed he was wearing a denim jacket at the time of the shooting- not a bulky one as previously described by a witness.
Giovanni da Silva, Mr Menezes' brother, called on the Metropolitan Police to "show all the video from the Underground so the world can see what they did to an innocent man".
Scotland Yard has confirmed that an offer of £15,000 - which it stressed was an "ex gratia" payment which would not affect any further compensation or legal action - had been made to the Menezes family.
The family rejected the offer, which is said to have caused them offence.
Their solicitor Gareth Peirce said the offer may have been a "deliberate attempt to ensnare families into inappropriate agreements or inappropriate decisions".
A memorial Mass will be held on Tuesday at 1930 BST at the Brazilian Chaplaincy in Westminster Diocese, St Anne's Church, east London, in memory of Mr Menezes.