The Lib Dem peer and campaign spokesman Lord Scriven claimed Mr Cameron had admitted privately to Nick Clegg that the Conservatives could not get the seats they needed to form a majority government.
His allegation drew an angry denial from Tory aides who insisted it was “100 per cent untrue” and showed that the Lib Dem campaign was in a state of panic.
Mr Clegg refused to be drawn directly on Lord Scriven’s comments but said there was “quite a gulf” between what Conservatives were saying publicly and privately, and that claims they could get an overall majority were a “big, fat fib”.
The latest row between the coalition partners was triggered by a tweet from Lord Scriven stating: “So Cameron has taken to lying on Tory majority. Nick Clegg told me that Cameron privately admitted to him that the Tories won’t win a majority.”
Appearing shortly afterwards on BBC news, the former Sheffield City Council leader said he spoke out after reading a newspaper report that the Tories were targeting Lib Dem-held marginals in an attempt to gain the 23 additional seats they need for a majority.
The article quoted Mr Cameron warning that a vote for the Lib Dems risked letting in a Labour government.
“If David Cameron is going to lie when he knows in his heart of hearts his private polls are showing that the British people are likely to give no party an overall majority, then so be it,” Lord Scriven said.
“I don’t take back anything I said in the tweet. People are not stupid. The opinion polls show what’s happening, the private opinion polls show what is happening.
“When I read that David Cameron is not telling the truth then I think that his private feelings do need to be made clear.”
A Conservative spokesman said: “This is 100 per cent definitely not true.
“The Prime Minister has repeatedly made clear we only need to win 23 seats to get a majority and he is totally focused on winning that majority.”
Campaigning in Twickenham – one of the Conservatives’ target seats – Mr Clegg said he was not prepared to comment on his private conversations with Mr Cameron.
However, he said: “All I would say to you is that the Conservatives are now trying to persuade people of something which is a big, fat fib – which is that they only need 23 seats to win a majority.
“They are not going to win a majority – they need 323 seats to win and they are not going to win, everybody knows that.
“I haven’t met a senior Conservative who, in private, will claim that they are going to win a majority.
“They know in private what everybody knows across the country, which is they are not going to win a majority.”
The Lib Dems said the Conservative claims they needed to win just 23 extra seats were a “con” as they failed to take account of constituency polling which suggested they would lose more than 50 seats they currently held.
But campaigning in the marginal Hastings and Rye constituency, Mr Cameron said voters were now faced with a clear choice between Conservatives and Labour.
“I think Nick Clegg is increasingly desperate because he knows the truth, which is there is an inescapable choice,” he said.
“On Thursday people have to choose – would they like me to continue as prime minister or do they want Ed Miliband and the SNP, the only other available option?
“My message is if you vote Lib Dem you are in danger of ending up with Ed Miliband, if you vote Ukip you are in danger of ending up with Ed Miliband.
“It’s only by voting Conservative you can guarantee that I’ll continue as Prime Minister, George Osborne will be back at his desk working on that long-term economic plan.”
The latest row between the coalition parties comes after Lib Dem Chief Secretary to the Treasury Danny Alexander last week released details of what he said were Conservative proposals to cut child benefit.
Labour leader Ed Miliband said that Mr Cameron was trying set England and Scotland against each other in an attempt to scare people into voting Conservative.
“He’s a desperate man because what he’s reduced to is trying to set this election up as a clash of two nations,” he said.
“He’s lost all the arguments: he’s lost the argument about the economy, he’s lost the argument about leadership, he’s lost the argument about his broken promises on the NHS, and so this is all he’s got left.”
Mr Miliband said that the alleged private admission “says it all” about Mr Cameron’s campaign.
“Other leaders, I gather, are conceding the outcome of this election,” he said during a campaign visit in Brighton.
“David Cameron appears to have now said he’s not going to win the election. I think that says it all about him and about his campaign.
“We are fighting every step of the way.”
Speaking on a visit to Wells, Mr Cameron was asked again if he had told Mr Clegg the Tories would not win a majority.
“Certainly not,” he said.
“Nick Clegg is obviously increasingly desperate because he knows he is losing seat after seat across the country.”
Mr Cameron added: “We won’t be put off by these increasingly desperate Liberal Democrat tactics.
“I could tell you all the things he has said to me in private. That would certainly surprise you.”
Asked about Lord Scriven’s claim, the Prime Minister said: “That’s complete nonsense and I think it’s rather desperate that the Liberals are saying things like this.
“The truth is, I think that they know that it comes down to a straight choice – do you want me to continue as Prime Minister or do you want Ed Miliband and the SNP in office?
“So, the pressure is on in Lib Dem-held seats because people know that if they vote Lib Dem they could end up with Ed Miliband and the SNP, not least because Nick Clegg said at the weekend that he would be happy to support that sort of government.”