Peter Hain insists he has been "open and clear" and will co-operate fully with an inquiry by the MPs' watchdog into his failure to declare donations.
Gordon Brown said Peter Hain would be a "great loss" to the cabinet
The work and pensions secretary is under increasing pressure after failing to register on time £103,000 given to his bid to be Labour's deputy leader.
Parliamentary standards commissioner John Lyon has confirmed he is to launch an inquiry into the funding row.
The Tories have stepped up their attacks on Mr Hain.
Officials at Mr Lyon's office say he believes there are grounds for an investigation, following a complaint by Tory MP David Davies.
Mr Hain could face suspension from the Commons if he is found to have flouted the rules on members' interests.
Prime Minister Gordon Brown has given his backing to the work and pensions secretary.
But he said his fate would be decided by the Electoral Commission and the Parliamentary Commissioner for Standards.
Speaking to The Sun newspaper, the prime minister said the minister was doing a "great job" and it would be a "great loss" if he had to leave the cabinet.
But, he added: "He took his eye off the ball and he has apologised. The matter must rest with the authorities, who will look at these matters."
Mr Hain blamed poor administration for his failure to declare the donations, saying it was "absurd" to suggest he attempted to hide anything.
He told the BBC: "A complaint has been made and I will co-operate fully with the relevant authorities to answer these complaints, of course I will.
"I've been open and clear about this from the beginning when I discovered these donations had been made late.
"I told the public, I told you in the media, I told the relevant authorities - the Parliamentary Commissioner for Standards and also the Electoral Commission.
"I'll continue to cooperate with everybody."
But Mr Hain faces questions over the role of a think-tank, the Progressive Policies Forum (PPF), in channelling donations.
The PPF employs no staff and has not published any research since it was set up in December 2006.
Shadow work and pensions secretary Chris Grayling has written to Mr Hain asking a number of detailed questions, particularly about the status of the PPF.
He said it was in the interests of all involved that the minister gave "immediate, clear and public answers" to put an end to the speculation.
The Electoral Commission is expected to produce its initial report into the funding row by the end of the week.
This will focus on whether the donors to Mr Hain were eligible to hand over the cash. A full report is expected later.
Chancellor Alistair Darling said: "The prime minister has made it clear that he values the work Peter Hain is doing at the Department of Work and Pensions. Clearly there is an inquiry and Peter has said that he will co-operate with that fully as everybody would expect."
But Labour MP Ian Gibson told BBC Radio 4's World At One he was surprised Mr Hain had allowed himself to get into "messes like this" and said he would resign in the minister's position.
"I think I would feel ashamed I had let the side down," he said.
He also questioned the use of the PPF to channel money to Mr Hain's campaign, adding: "It gives think-tanks a bad name."
But fellow Labour MP Paul Flynn said "innuendos" about the PPF were wrong and there was "nothing sinister" about it - as it had been set up specifically to support Mr Hain and his policies.
George Osborne is a close friend of Tory leader David Cameron
He said a "rap across the knuckles" was more appropriate than asking Mr Hain to abandon his career, saying that would be a "terrible shame".
The Tories have themselves come under fire after it was reported that shadow chancellor George Osborne did not declare £487,000 of donations to the Commons Register of Members' Interests.
Earlier, Conservative leader David Cameron said the two cases were "completely different" because Mr Osborne's donations had been registered with the Electoral Commission.
Of Mr Hain, he said: "He has a choice: he can get out there and explain himself or I think he will have to leave the cabinet.
"It is quite a contrast: George Osborne on the Today programme this morning answering all the questions about the funding of his office - Peter Hain nowhere to be seen, and I do not think that is acceptable."