An opinion poll did little to ease the pressure on the Prime Minister when it suggested nearly one-third of voters who backed the Tories at the last General Election now think UKIP, the anti-EU party, has the best policies on Europe.
The Tories were beaten into third place in the Eastleigh by-election behind UKIP.
Liberal Democrat candidate and former Hampshire councillor Mike Thornton won. He was welcomed to Westminster by party leader and Deputy Prime Minister Nick Clegg yesterday.
Meanwhile, in his speech, at an undisclosed location in northern England on Thursday, Mr Cameron is once again expected to insist Britain must stick to the Coalition Government's austerity programme.
Last week, he told MPs the loss of the UK's triple A credit rating meant the Coalition had to cut the nation's budget deficit further and faster.
Tory backbenchers have upped the pressure on Chancellor George Osborne by warning Britain is on track to become a "basket case" unless tougher action is taken on the deficit.
This comes amid increased squabbling within the UK Government over spending plans with Philip Hammond, the Defence Secretary, going public with a call for welfare to bear a greater share of austerity pain.
However, No 10 yesterday suggested benefits had been cut enough as the PM's spokesman pointed out that in his Autumn Statement the Chancellor had announced an extra £3.6 billion of cuts to the welfare budget.
He added: "If new and specific proposals were to emerge, then they would need to be considered."
The spokesman also sought to play down open ministerial squabbling on where Mr Osborne's axe might fall in further cuts, saying: "One would always expect secretaries of state and departmental ministers to make a robust case on behalf of their priorities. That is entirely to be expected."
Lord Forsyth, the former Conservative Scottish Secretary, said it was time the Chancellor started cutting taxes to encourage investment.
However, Vince Cable, the Liberal Democrat Business Secretary, said that while he was all for encouraging investment, there was "not much scope for tax-cutting".
Meanwhile, Tory backbencher Kwasi Kwarteng, speaking at an event organised by the Institute of Economic Affairs think-tank, floated the idea of tax breaks for new business start-ups. "If you look at what is happening in America, they are actually cutting spending in a very hands-on, engaged way.
"They do what it takes. Unfortunately, we have not seen massive spending reductions in Britain," he said.
Mr Kwarteng wants to see foreign aid spending slashed by one-third and floated measures such as £1bn of tax breaks for new business start-ups.