Extracts from a speech that Cameron was planning to make this morning show that the prime minister intended to make clear that he wants the UK to play a "committed and active" part in the EU in future.
But he was also planning to warn that, if changes are not made to address the three key challenges of eurozone crisis, economic competitiveness and dramatically declining public support, "the danger is that Europe will fail and the British people will drift towards the exit".
But the extracts released by Downing Street did not reveal whether the prime minister intended to commit himself to an in/out referendum on British membership of the EU following the renegotiation of its terms which he has already said he plans to undertake after the 2015 general election.
The prime minister's reference to possible exit from the EU is his starkest warning yet of the consequence of failure to reform Europe, and echoes Chancellor George Osborne's comment to a German newspaper last week that "in order that we can remain in the European Union, the EU must change".
Cameron's long-awaited speech in the Netherlands was dramatically postponed last night, after the prime minister decided he should stay in London to deal with the bloody fallout from the hostage crisis in Algeria.
The postponement - to a date yet to be fixed - is the latest mishap to befall the his attempt, several months in the planning, to spell out his vision for the future of Europe.
First trailed as early as September, the speech has already been so delayed that Cameron joked he was taking a "tantric" approach to it.
He prepared the ground for a public statement of his Europe policy by speaking to opposite numbers in Germany and Sweden in the Netherlands over the last few days, and discussed last night what he planned to say with French president Francois Hollande and US president Barack Obama.
Segments of the speech released last night showed that he was intending to tell Europe's leaders that they have a "duty" to respond to the growing frustration of their people and to accept change to the way the EU operates.
"More of the same" would condemn EU nations to "less competitiveness, less growth, fewer jobs", he argued.
According to the speech extracts, Cameron was planning to use the address to set out what he termed a "positive vision for the future of the European Union. A future in which Britain wants, and should want, to play a committed and active part".
He was planning to tell an audience of business people and EU diplomats that Europe "must change, both to deliver prosperity and to retain the support of its peoples".
The woes of the euro were already forcing "fundamental change" on the structures of the EU, while a "crisis" in competitiveness was allowing emerging nations around the world to "soar ahead" of Europe, he was due to say.
And he was planning to warn that "there is a gap between the EU and its citizens which has grown dramatically in recent years and which represents a lack of democratic accountability and consent that is - yes - felt particularly acutely in Britain".
The speech extracts warned: "If we don't address these challenges, the danger is that Europe will fail and the British people will drift towards the exit.
"I do not want that to happen. I want the European Union to be a success and I want a relationship between Britain and the EU that keeps us in it."
According to Cameron, the austerity measures taken to deal with the crisis in the eurozone have given added urgency to the issue of the EU's democratic legitimacy.
"People are increasingly frustrated that decisions taken further and further away from them mean their living standards are slashed through enforced austerity or their taxes are used to bail out governments on the other side of the continent," he was intending to argue.
"And yes, of course, we are seeing this frustration with the EU very dramatically in Britain. Europe's leaders have a duty to hear these concerns. And we have a duty to act on them."
In a riposte to supporters of the status quo in Europe, Cameron was planning to warn: "More of the same will not secure a long-term future for the eurozone. More of the same will not see the European Union keeping pace with the new powerhouse economies. More of the same will not bring the European Union any closer to its citizens.
"More of the same will just produce more of the same - less competitiveness, less growth, fewer jobs. And that will make our countries weaker, not stronger."
The Prime Minister's approach won the support of London mayor Boris Johnson, who said in a speech last night that he believed voters would back a renegotiated membership which saw Britain stay in the single market but remove some of the "barnacles" which have grown up around EU membership.
There was also renewed US pressure, with President Barack Obama telling Cameron that Washington greatly valued Britain being in the EU.
Cameron set out his proposed speech in a call with the president yesterday afternoon before it was postponed.
The White House made clear that Obama maintained the pressure on the Prime Minister in the call.
A spokesman said: "The president underscored our close alliance with the United Kingdom and said that the United States values a strong UK in a strong European Union, which makes critical contributions to peace, prosperity, and security in Europe and around the world."