No 10 says Mr McBride's e-mails were "juvenile" exchanges between friends
Gordon Brown is facing pressure from the Conservatives and some in Labour to distance himself further from e-mails which discussed smearing senior Tories.
Damian McBride quit as a No 10 aide after his unfounded claims about Tory leader David Cameron and shadow chancellor George Osborne became known.
Mr Cameron has demanded a public apology from the prime minister.
A No 10 spokesman has said no-one else in Downing Street knew about the "juvenile and inappropriate" messages.
But the Conservatives say that is not a good enough explanation.
'Into the gutter'
They have demanded an inquiry as well as a guarantee that such material will never again be written in Downing Street.
The Tory leader was "absolutely furious" about the e-mails, his spokeswoman said.
BBC political correspondent Laura Kuenssberg said there were rumblings among Labour backbenchers, with some urging Mr Brown to do more to distance the party from the scandal.
The stables do need to be cleaned out and it goes beyond the two particular oiks we are obsessed with at the moment
Tony Blair's former economic advisor
Downing Street has already said it was Mr Brown's view that there was "no place in politics for the dissemination or publication of material of this kind".
Tony Blair's former economic adviser Derek Scott told BBC Radio 4's Today programme that although he did not think Mr Brown was directly responsible, the smear tactics represented a culture that came from the top.
He said: "I think the tone, whether in a business or a political party, is set from above."
"The really damaging thing is they think they are being ultra loyal to Mr Brown and the Labour Party, but in fact they are doing the Labour Party an awful lot of damage.
"The stables do need to be cleaned out and it goes beyond the two particular oiks we are obsessed with at the moment," he added.
Chris Grayling: "This was right at the very heart of our government"
Cabinet Office Minister Liam Byrne has said Mr Brown knew nothing of the e-mails.
But Labour backbencher John McDonnell has called on Mr Brown to act decisively by launching an independent inquiry into who was involved.
"Smear tactics like this are not the Labour way," he said.
"They drag the Labour Party into the gutter. They just add further to the undermining of the belief that Labour Party supporters have placed in our party."
The e-mails were originally sent in January by Mr McBride to former government spin doctor Derek Draper, who runs the LabourList blog and was proposing to set up Red Rag, a new gossip-led site.
However, they came to the attention of Paul Staines, author of the "anti-politics" Guido Fawkes blog, who revealed their existence.
Charles Clarke, a former home secretary, said the resignation was the end of the "Damian McBride issue", but did not address the wider problem of smear campaigns in politics.
There is tangible anger in some quarters of the party - concern that the publication of the e-mail extracts smears Labour in general
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.