"We would have hoped that Gordon Brown would have said that he was looking into this, that due process was being followed and that he takes these issues seriously," said Pratt, who runs the National Bullying Helpline.
On Monday, Professor Cary Cooper, a patron of the charity, resigned saying that Pratt had breached client confidentiality with her remarks.
David Cameron, the Conservative leader, said: "They are very serious matters and I'm sure that Number 10 Downing Street and the civil service in some way will want to have some sort of inquiry to try and get to the bottom of what has happened here."
Peter Mandelson, the country's business secretary, said the prime minister appeared to be the victim of a political campaign and dismissed calls for an inquiry.
"Do you not think we have better things to do than to chase up every rag, tag and bobtail rumour, innuendo and smear that anyone with a book to sell chooses to pump into the ether" Mandelson said at a conference in London.
"We've got a country to run, that's what's important to us. Nobody bullies, nobody tolerates bullying in this government, in any part of this government, period, zero, and that's it, OK?"
Brown, 59, replaced Tony Blair in mid-term in 2007 after serving as his finance minister for a decade.
|Cameron said the accusations were a 'serious matter' [EPA]
He is often portrayed as intense and brooding and some critics, even within his own party, say he is an electoral liability.
The prime minister has recently opened up more in interviews, showing a more emotional side when discussing the death of his new-born daughter in a television chat show a week ago.
The allegations against Brown overshadowed a poll published in The Sunday Times newspaper which showed the ruling Labour Party, in power since 1997, had narrowed the gap on the Conservatives to six points
Such a margin could lead to a hung parliament in which no party has an absolute majority.