Their calls were sparked by claims from the founder of an anti-bullying charity that her helpline had received calls from Downing Street staff complaining of the way they were treated.
But First Secretary Lord Mandelson dismissed calls for an inquiry, claiming they were part of a "political operation" to undermine Prime Minister Gordon Brown.
Lord Mandelson accused Conservative press officers of guiding journalists towards National Bullying Helpline founder Christine Pratt in the assumption that she had "some fuel to throw on this fire".
But he was accused in turn by the Conservatives of trying to "smear" Mrs Pratt by wrongly portraying her as a political stooge.
Downing Street insisted that there was a "zero tolerance" approach to bullying in the civil service, and that there were no grievance procedures under way in relation to Number 10 or the Cabinet Office at present.
Mrs Pratt spoke out following the publication of extracts of a book by Andrew Rawnsley that included accounts of Mr Brown pulling a secretary from her chair, "roughly shoving" an aide, and four-letter-word rants that frightened staff.
She said she "saw red" after Lord Mandelson issued a categorical denial of bullying in Mr Brown's office.
The helpline had received two calls from the Deputy Prime Minister's Office and two or more from Number 10, she said.
But she acknowledged that none referred to claims of bullying by Mr Brown himself.
"Absolutely not, and nor have we said that Gordon Brown is a bully," she told BBC Radio 4's Today programme. "Our concern here is the public statement from No 10 of denial ... we would just want Gordon Brown and No 10 to lead by example."
Mrs Pratt's decision to go public with the complaints sparked protests from other anti-bullying campaigners.
Professor Cary Cooper, a celebrated expert on workplace stress, quit as a patron of the National Bullying Helpline, saying: "I am resigning now on the grounds that I think she breached confidentiality.
"One of the things that is really important for any helpline or any counselling service is to retain confidentiality of the people calling up."
And the director of the charity Bullying UK, Liz Carnell, said she was "horrified" by Mrs Pratt's move and was complaining to the Charity Commission.
"Gordon Brown's office is small and the National Bullying Helpline's comment will almost certainly identify this person who turned to the helpline in despair," said Ms Carnell.
"We suggest Mrs Pratt considers her position, given the damage she has caused to the anti-bullying sector, where confidentiality is paramount."
The Charity Commission said it had received a number of complaints about the National Bullying Helpline over the weekend and was considering them to see whether they fell into its remit.
"We will be contacting the charity for further information," said the Commission in a statement.
Mr Cameron called today for an inquiry into the bullying allegations by Sir Philip Mawer, who polices the ministerial code.
Speaking at an event in London, the Tory leader said: "These are very serious matters. I'm sure that Number 10 Downing Street and the civil service in some way will want to have some sort of inquiry to get to the bottom of what has happened here.
"To me, all of this just says we need to get on, have a general election and have a fresh start."
And Mr Clegg called on Downing Street to clear up exactly what had been going on behind closed doors at Number 10.
"The allegations that junior staff, voiceless staff, have been going to a national bullying hotline seems to me to be very important," he told the BBC Radio 4 Today programme.
"Bullying is wrong, it is totally out of order in whatever walk of life you work, however important you are.
"In politics, how you behave behind closed doors - even the great closed doors of Number 10 - how you behave once the cameras are switched off, does kind of matter, so I do think this does need to be cleared up by Number 10."
Lord Mandelson said that he had initially viewed the row as a "storm in a teacup".
But speaking to reporters at an investment conference in London, he added: "It now looks like more of a political operation that's under way directed at the Prime Minister personally...
"I even gather that Conservative Party press officers were active yesterday, guiding journalists towards Mrs Pratt, assuming that she had some fuel to throw on this fire."
In response, a Conservative Party spokesman said: "This is the default position of the Downing Street machine - to smear the messenger as they have done so many times before.
"Serious allegations of bullying within Downing Street have been made by both Andrew Rawnsley and Mrs Pratt. Instead of addressing them, they are dismissing Mrs Pratt as a Tory party stooge. She is not.
"Now Peter Mandelson claims that Mrs Pratt was 'guided by some Tory Party press officer'. She was not.
"There is a simple way for Gordon Brown to clear up these serious allegations, that is by instructing Sir Phillip Mawer to investigate whether any breach of the Ministerial Code has taken place."
Lord Mandelson said there had been no actual evidence of bullying at Number 10 and that it would not be tolerated if it were to take place.
"There is zero tolerance for bullying in the Government as a whole, and certainly at the centre of government," he said.
"I just think it's really, really unfortunate that somebody should use a charity in this way... You do have to ask yourself why she isn't more sure of her ground, why she isn't clear in the allegations she is making, and why she isn't able to produce more evidence for the points that she's making."
Mrs Pratt said she felt "slightly intimidated" by some of the reaction she had received to her allegations, including an email from former Home Secretary Jacqui Smith accusing her of having "comprehensively breached" her duty of confidentiality towards callers.
Speaking on GMTV, Mrs Pratt said: "We haven't breached confidentiality and we wouldn't do that."
She added: "There is no political edge at all. We are a non-political organisation."
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