He was speaking ahead of an MPs debate on giving ministers emergency powers to take the bank into "temporary public ownership".
Mr Darling, who has so far resisted calls for his resignation, will pilot through legislation allowing the stricken mortgage lender to be nationalised.
But speaking on GMTV today, Mr Cameron said: "When you've got a Chancellor who I think everyone would accept has lost all credibility, what I think is right is for the Prime Minister to reconstruct his government."
He continued: "At the end of the week, if I was the Prime Minister, I would say to my Chancellor that he has to move from his job and that a new Chancellor is required to take the country forward and help restore economic credibility."
Yesterday, shadow chancellor George Osborne branded Mr Darling "politically a dead man walking", while shareholders have threatened legal action.
Northern Rock employees are also braced for thousands of job losses as the business's operations are scaled back.
Justifying the nationalisation decision to the Commons yesterday, Mr Darling insisted: "We could have let the bank go under. But the risks to the wider financial system, for savers and the general public were not acceptable."
Gordon Brown, asked directly at a No 10 news conference whether the bank would return to private ownership before the next election, not due until May 2010, replied: "We can't have a timetable when we are talking about the return of better market conditions as a first step to resolving the issue."
Ron Sandler, the new Treasury-appointed chief of Northern Rock, told a press conference in Newcastle he believed it would be "some years" before billions in government loans and guarantees were repaid in full.
Period
He said: "I understand that other plans put forward spoke of a period of some years before the loans could be properly repaid and the bank returned to a situation where the government guarantees had disappeared.
"I do think it's clearly unrealistic to be talking about months - we are clearly talking about a period of some years."
He said he wanted to give the bank "a period of stability" and not "run it down to extinction".
But both he and the Chancellor acknowledged that restructuring the bank now had to be done under tight competition rules and EU guidelines designed to ensure state aid did not distort financial markets.
And Mr Sandler refused to speculate about possible job losses among the 6,500 staff.
The Conservatives have also demanded the publication of advice given to the Government by its City advisor Goldman Sachs regarding Northern Rock's potential sale.
Treasury Chief Secretary Yvette Cooper last night refused to confirm reports that taxpayers faced a £100m bill for advice on the crisis from City lawyers and bankers.
But she defended the Government's need to take "serious legal advice" and accepted that large sums would be indirectly borne by the public purse through Northern Rock.
"I can't tell you what the figure is," she told BBC 2's Newsnight. "But what I can tell you is that when a bank gets itself into trouble, inevitably there are fees and costs involved."
The Banking (Special Provisions) Bill is expected to clear Parliament by the end of the week.
It does not make any explicit reference to Northern Rock, instead empowering ministers to take control of financial institutions "in certain circumstances" where it is necessary to protect "the stability of the UK financial system". Tweet

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