Mr Duncan Smith has warned the government risks dividing society with politically-motivated spending cuts.
Mr Cameron is to reject this - and No 10 has rubbished claims of a rift with George Osborne, saying the chancellor still has the PM's full confidence.
The disability cuts Mr Duncan Smith quit over will be shelved.
Downing Street said the changes to Personal Independence Payments (PIP) announced ahead of Mr Osborne's Budget last week would not go ahead "in their current form".
But alternative proposals for saving the Â£4bn earmarked for the savings would not come until the Autumn Statement towards the end of the year.
Labour leader Jeremy Corbyn has called for Mr Osborne's resignation and Conservative MPs have spoken out against the leadership with Mr Duncan Smith's former ministerial team divided in their responses to his resignation.
Treasury minister David Gauke is answering an urgent Commons question from Labour's shadow chancellor John McDonnell on changes to the Budget - Mr McDonnell had called for the chancellor himself but this appears to have been ignored.
This will be followed by a statement from Mr Cameron - ostensibly on last week's EU summit - at which he is expected to restate his commitment to "compassionate Conservatism" and reject Mr Duncan Smith's criticisms of his style of government.
Former Tory leader Lord Howard urged MPs to "listen to what the prime minister has to say" and to "calm down".
Mr Duncan Smith set out the reasons for his surprise resignation in an interview with the BBC's Andrew Marr, in which he said the way Mr Osborne had cut benefits in his Budget at the same time as cutting taxes for the better off was "deeply unfair" and that he had become "semi detached" from government.
Ross Hawkins, BBC political correspondent
George Osborne: chief tactician and patron, a man whose word makes or wrecks careers.
That was the view of many Tory MPs for a very long time. Now, many doubt he will ever be their leader, or even the chancellor much longer.
It's not that they think David Cameron is poised to sack him or that he's about to resign, as Labour demand.
They simply believe a swift leadership election is highly likely whatever the result of 23 June's EU referendum.
If it does come that soon, there will be - one predicts - a "genocide of the Cameroons and Osbornites".
A stubbornly enduring deficit, a tax credit U-turn, and the sheer numbers of MPs who have chosen to back a leave vote at the referendum have seen Osborne's authority leak.
A weekend of melodrama has - in one Tory MP's view - burst the dam.
Read more from Ross
Lord Howard played down Mr Duncan Smith's criticisms of government policy - but Conservative MP Sarah Wollaston said they were "very serious".
"Today, when David Cameron stands up, he has to reaffirm the message that led many people like myself to join the Conservative Party in the first place when he became leader," she said.
"Are we about social justice? Are we about spreading the burden fairly? We need to hear that very clear message today."
She suggested pensioner benefits - which the Conservatives pledged to protect in their manifesto - should be cut to make up the shortfall.
A number of senior Conservative figures have questioned Mr Osborne's credentials to replace David Cameron as prime minister when he steps down.
Former chief whip Andrew Mitchell described Mr Osborne's abortive attempt to reform PIP as a "cock-up".
He said Mr Osborne was "not the only candidate" for the party leadership and there were a "large number" of alternatives.
Influential backbencher David Davis told the BBC's Victoria Derbyshire the Â£4bn welfare savings should be cancelled and suggested Mr Osborne should be moved to another department if he wants to be a successful replacement for David Cameron as prime minister.
London Mayor Boris Johnson, seen as Mr Osborne's main rival for the top job, is returning from a skiing holiday and has yet to comment.
BBC assistant political editor Norman Smith said it was understood Mr Cameron had urged Mr Osborne to avoid any major controversy in the Budget so as to avoid fuelling discontent among Tory MPs ahead of the EU referendum.
Despite this, our correspondent said, Downing Street insists "the two men remain as close as ever", and have dismissed reports that the prime minister will seek to distance himself from Mr Osborne.
Number 10 has stressed that PIP will still have to be reformed in the future as the cost is "unsustainable."
Mr Corbyn told the BBC Mr Osborne should be "considering his position".
"His Budget simply doesn't add up and it unravelled within hours of him presenting it. This isn't the first time a George Osborne Budget has unravelled," the Labour leader told BBC1's Breakfast programme.
"It seems to me we need to look at the very heart of this government, at its incompetence, at the way it puts forward proposals that simply don't add up and expects the most needy in our society to take the hit for them."
Who would the disability changes impact?
Prime Minister David Cameron will later attempt to halt the civil war in his party caused by Iain Duncan Smith's resignation from the cabinet.