On Monday it emerged at the end of a year-long terror trial that MI5 had two of the 7 July bombers under surveillance a year before the attacks.
Ministers are opposed to an inquiry but a parliamentary committee will consider why the bombers were not picked up.
Those affected by the 2005 attacks have delivered a letter to the Home Office.
The document, requesting an "impartial public inquiry", was handed to an official from the department.
BUGGED BY MI5
It says one of the purposes would be "to examine issues aimed at saving lives, minimising suffering and improving the response of government agencies to the continuing threat of terrorist attacks".
The letter was signed by more than 18 people, including 7 July survivors Paul Mitchell and Jacqui Putnam, as well as relatives of those affected - such as Ros Morley, whose husband, Colin, was killed.
Survivor Rachel North said: "This is not about blame but about future public safety - understanding what happened, how it happened and to stop it happening again."
In response, the Home Office released a statement saying the home secretary would "give very careful consideration" to the letter.
The statement added: "However, as we have consistently maintained, experience has shown that a fuller public inquiry can take years and divert huge resources."
On Monday five men were given life sentences for a foiled plot to build a huge fertiliser bomb for a UK attack.
It emerged during the trial that MI5 had tailed London suicide bombers Mohammad Sidique Khan and Shehzad Tanweer while investigating that case, but took no action.
The BBC has now published a full transcript of an MI5 bugged conversation in which the bomber discusses leaving the UK to join jihadi extremists in the Pakistan-Afghanistan border areas.
One issue being looked at is what MI5 told both the public and politicians in the wake of the 7 July attacks.
EVIDENCE SINCE TRIAL
The media were briefed that Khan and fellow bombers were "clean skins" - men with no previous record of terrorist associations.
But evidence following the end of the trial reveals MI5 photographed Khan as he met other extremists, followed him home - and by the summer of 2004 they knew his surname and that he owned a car.
Danny Biddle, who survived the 7 July attacks, said the revelation that there were links between the suicide bombers and those behind the fertiliser plot meant a public inquiry was now essential.
"This is about finding out how this could be allowed to happen and how nobody could stop it. That needs to be investigated and to totally dismiss a public inquiry is shameful."
Paul Dadge, who also survived the London bombings, said he believed the attacks may have been prevented if the leads had been followed, but argued that it was important to praise the security services for securing convictions.
Prime Minister Tony Blair has asked the parliamentary Intelligence and Security Committee (ISC) to consider why the 7 July bombers were not picked up.
The committee, which comprises MPs and Lords, is expected to examine claims that West Yorkshire Police special branch was not told about the MI5 surveillance operation.
ISC chairman Paul Murphy MP has previously indicated that police were informed.
Former Tory Defence Secretary Lord King cast doubt on the impartiality of the committee, saying it should be led by a member of the Opposition to ensure independence.
Mr Blair rejected calls for a public inquiry, although he said he "totally" understood why some people sought one.
He told GMTV: "The problem if you have an independent public inquiry into something like this is you will divert all their energy and attention into trying to answer the questions that come up in the inquiry."
WHAT MI5 TOLD COMMITTEE
The revelation that one of the 7 July bombers met up with one of the fertiliser bomb plotters - Omar Khyam - at a terrorist training camp in Pakistan has caused concern.
However, the head of Pakistan's National Crisis Management Centre, Brigadier Javed Iqbal Cheema, said anyone "who spends a lot of money and travels to Pakistan...[is] already motivated for a particular reason".
The Liberal Democrats and Conservatives have both called for an independent inquiry into the 2005 London bombings.
^ Back to top | BBC Sport Home | BBC Homepage | Contact us | Help | ©
~RS~q~RS~~RS~z~RS~43~RS~)