Mr Osborne strenuously denies attempting to solicit a donation
Gordon Brown's call for an inquiry into shadow chancellor George Osborne is a "desperate attempt to keep the story going", the Conservatives have said.
Mr Brown called claims Mr Osborne had attempted to solicit a £50,000 donation for the Tories from a Russian billionaire a "very serious matter".
He added that they should be investigated by the "authorities".
But a Conservative spokesman said the allegations had already been "dealt with" in a statement by Mr Osborne.
Tory defence spokesman Liam Fox told Sky News that Mr Osborne had "no case to answer" over the matter.
Financier Nathaniel Rothschild has accused the shadow chancellor - and Tory chief executive Andrew Feldman - of attempting to solicit a £50,000 donation for the Tories from Russian businessman Oleg Deripaska while on holiday in Corfu during August.
Mr Osborne and Mr Feldman strenuously deny doing so, saying that the issue was raised by Mr Rothschild.
But the financier says he has a witness - multimillionaire hedge fund manager James Goodwin - to a conversation he and Mr Osborne had about the matter, and has let it be known that he would go to court over the matter.
It is illegal for someone not on the UK electoral register to make a donation to a political party.
During prime minister's questions on Wednesday, Mr Brown said: "This is a very serious matter indeed and I hope it is investigated by the authorities."
Pressed later about which allegations Mr Brown had been referring to, Downing Street said those made "over the past 24 hours" but would give no more details.
A spokesman also refused to be drawn on which "authorities" would carry out an investigation, saying "whichever authorities are appropriate".
Labour MP Denis MacShane has written to Mr Osborne, warning that "it is likely there will be issues for the Electoral Commission or other authorities to investigate, but I will give you the opportunity to answer the following questions before deciding whether to refer the matter to them".
'Not an offence'
He asked a series of questions, including whether "you had a conversation with Mr Rothschild, witnessed by Mr James Goodwin, about a donation to the Conservative Party from Mr Deripaska".
However, the Electoral Commission said it would not launch an inquiry, as "soliciting a donation is not an offence".
Business Secretary Lord Mandelson, who was then European trade commissioner, stayed on Mr Deripaska's yacht, moored off Corfu, at the same time as Mr Osborne stayed at Mr Rothschild's villa.
A Conservative spokesman said: "The prime minister's statement in the Commons and Denis MacShane's letter are a desperate attempt to keep this story going.
"All the questions raised in Denis Macshane's letter are dealt with in yesterday's statement from George Osborne and Andrew Feldman.
"The person who hasn't answered any questions is Peter Mandelson, who should immediately set out a detailed account of his numerous dealings with Mr Deripaska."