The prime minister is facing calls to sack his spokesman for referring to government weapons inspector Dr David Kelly as a "Walter Mitty" character.
Dr Kelly's funeral will be held on Wednesday
Tom Kelly made the remarks during what Downing Street regarded as off-the-record conversations with journalists about the Hutton inquiry into the scientist's death.
On Tuesday, Mr Kelly said he "deeply regretted" making the comments and "apologised unreservedly" to Dr Kelly's widow and family.
But his remarks prompted anger in Westminster, with Labour MP Glenda Jackson saying the briefing was "unspeakable" and demanding that Mr Kelly is sacked.
"Number 10's capacity to disgust us would seem positively boundless," she told BBC Radio 4's Today programme.
"In my view, [Tom Kelly] should lose his job. I don't think he should be afforded the luxury of resigning - I think he should be sacked."
Dr Kelly's friend Professor Alistair Hay said the remarks were "heartless in the extreme".
He added: "I'm very surprised by the whole thing. It's deeply shaming."
Liberal Democrat leader Charles Kennedy said Mr Kelly's position had now become "extremely difficult".
But it should be up to the cabinet secretary, not politicians to look at the civil servant's future.
Mr Kennedy told BBC Radio 4's PM programme in most occasions an unqualified apology would have settled the row.
But there was this "rather curious, almost amoral view that it's alright for civil servants... to cast aspersions of this type on someone who is not yet in their grave as long as it remains private", he added.
Mr Kelly, one of Mr Blair's two official spokesmen, was named on Tuesday as the "senior Whitehall source" who had suggested Dr Kelly had misled government officials and the BBC.
The "Walter Mitty" description refers to a character who fantasised about being a hero in a story by US author James Thurber.
In a statement, Mr Kelly said it had been his public and private view that the family of Dr Kelly "should be left to grieve" and the Hutton inquiry be allowed to complete its investigation.
"I deeply regret, therefore, that what I thought was a private conversation with a journalist last week has led to further public controversy.
"That was not my intention, nor, most emphatically was a I signalling a government strategy aimed at discrediting Dr Kelly, as I have explained to the deputy prime minister."
Mr Kelly said he had been trying to outline the questions that the Hutton Inquiry would have to address.
"It was in that context that the phrase 'Walter Mitty' was used, but it was meant as one of several questions facing all parties, not as a definitive statement of my view, or that of the government.
"I therefore unreservedly apologise to Dr Kelly's widow and her family for having intruded on their grief."
Deputy Prime Minister John Prescott on Tuesday sent the spokesman's statement to Dr Kelly's widow and has written to her expressing the government's deep regret and apologies.
Downing Street had on Monday tried to distance itself from the Walter Mitty comments.
But later in the day, Number 10 acknowledged that someone there had spoken to the Independent, but said the conversation did not reflect the government's view.
In a separate statement on Monday, Mr Prescott said: "I trust that no one in government would comment on Dr Kelly at such a
sensitive time, before the funeral and while the Hutton inquiry is under way."
Mr Prescott is due to stand in for the prime minister, who is on holiday, at Dr Kelly's funeral on Wednesday.
Labour MP Kevin Brennan said he believed that Cabinet Secretary Sir Andrew Turnbull should interview Mr
Kelly about his actions to see if he had breached the Civil Service Code - but Sir Andrew's office insisted it was an "internal affair".
"The Secret Life of Walter Mitty" was written by US author James Thurber
Story described as "a fantasy and social criticism about an American daydreamer"
First published in 1941
Mitty defined in US dictionary as someone "who indulges in fantastic daydreams of personal triumphs"
Dr Kelly is thought to have committed suicide after speculation - later confirmed by the BBC - he was the source of stories that raised concerns over the way the government presented its case for war with Iraq.
Lord Hutton, whose inquiry will begin taking evidence on Monday, has announced that the probe will not be televised.