With nominations to open later, Work and Pensions Secretary Stephen Crabb has said he will seek the leadership, promising a "One Nation" Conservatism.
Boris Johnson is reported to have the support of 100 MPs, and Home Secretary Theresa May is also expected to stand.
Education Secretary Nicky Morgan, who could also declare, said the "positive case" for immigration should be made.
Nominations for the contest will open on Wednesday, with candidates then having 24 hours to come forward.
The leadership race was triggered by Mr Cameron's decision to stand down after the UK voted to leave the EU last week.
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All the prospective contenders are gauging support within the parliamentary party, which will whittle down the candidates to two in a series of ballots before Conservative party members decide between them. The final result is expected on 9 September.
The BBC's political correspondent Carole Walker said the key issue in the campaign would be who could unify the party after the bruising EU referendum and set out a credible plan for a Brexit after Mr Cameron said he would leave the fraught and complex negotiations to his successor.
Mr Crabb is expected to stress his "One Nation" credentials and his desire to work with leaders in Wales and Scotland as he teams up with Business Secretary Sajid Javid on what has been described as a "dream ticket".
"We will enact the British people's wishes on the EU," he wrote in an article in the Daily Telegraph setting out his agenda.
"The verdict was clear; there is no going back. A second referendum is out of the question. What the country needs now is a clear direction, not further instability."
He said Mr Javid has "unrivalled" experience to leading Brexit negotiations and "to secure a deal which protects UK business and workers".
Youth appeal
Mr Crabb also said he would listen to the public's concerns. "This referendum has cast a harsh light on the deep economic and social divisions that still blight our country," he said.
"People are stuck on low wages - even no wages - despairing at their hollowed-out town centres and a sense of being left behind in a fast-changing world.
"So my government will confront it, spreading economic and social opportunities to every corner of Britain in a way we haven't seen before."
Ms Morgan told the same newspaper that negotiations must take into account the interests of the 48% of the population who voted to remain in the EU as much as the 52% who wanted to leave.
In a pitch to younger voters, who she says have been left "disillusioned" by Brexit, the MP - who is taking soundings on a potential bid - said the Conservatives must engage with "21st Century Britain as it is not a nostalgic vision of how we might wish it to be".
"It is young people who are must vulnerable to the economic shocks that will follow the vote to leave," she wrote.
She said there was "a recipe for mass disillusionment among the young".
"The Conservative Party must make absolutely clear that it is committed to fairness across the generations with a clear platform aimed at young people on issues such as education, housing and jobs."
Mr Johnson is regarded by many as the favourite at this stage, having been a leading figure in the victorious Leave campaign and being popular among Conservative activists who will make the final choice.
Having already won the backing of Justice Secretary Michael Gove, Mr Johnson is expected to announce he has got the endorsement of Environment Secretary Liz Truss, a prominent Remain campaigner. The Sun suggested he had won the backing of 100 MPs already.
Mr Johnson is expected to be advised by Lynton Crosby, the Australian strategist behind the Tories' last general election campaign.
Also considering standing is John Baron - a backbench Eurosceptic who led the initial drive to hold an EU referendum. Health Secretary Jeremy Hunt and Energy Minister Andrea Leadsom are also weighing up leadership bids.
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