Cameron said Britain was not a "dispassionate observer" of events in the EU, where "everything was changing and fears about Europe's economic future were "intense".
He made it clear, although he described himself as a eurosceptic, that Britain would remain in the EU, and that his country had "a profound national interest" in ensuring the resolution of the eurozone crisis.
Historically, the EU has been an "effective anchor for democracy and prosperity", Cameron said at the Lord Mayor's Banquet in London. But today it had become "out of touch" with its citizens, with "pointless rules and regulations that stifle growth, not unleash it", he said.
"It does not have to be like this. Out of crisis can come opportunity for the European Union, if its member states are ready to grasp it," Cameron said. "We’ve a right to ask what the European Union should and shouldn’t do… and change it accordingly."
He described the Union he was looking for in the following terms:
"One that is outward-looking – with its eyes to the world not gazing inwards.
"One with the flexibility of a network, not the rigidity of a bloc – whose institutions help by connecting and strengthening its members to thrive in a vibrant world, rather than holding them back.
"One that understands and values national identity and sees the diversity of Europe’s nations as source of strength."
Cameron said he would seize the opportunity "for powers to ebb back instead of flow away", and "for the European Union to focus on what really matters".
Germany seeks changes
Cameron's statements are not entirely new and come against the backdrop of calls by Germany for changes in EU treaties, needed to enforce the economic governance inside the eurozone.
The UK would seek EU opt-outs on directives affecting labour rights and financial services regulation if eurozone countries adopt fundamental treaty changes, Cameron told the UK Parliament on 6 September.
Two weeks ago, Cameron said the Commission had a duty to safeguard the 27-member EU bloc, amid concerns of 'caucusing' - decision-making by some members of a group - by the 17 eurozone states.
Some on the right of Cameron's Conservative party have called for Britain to leave the EU or at least to use the eurozone crisis as an opportunity to claw back powers from Brussels.
Cameron faced a rebellion when more than a quarter of his members of parliament defied him last month and demanded a referendum on EU membership. Eventually, the Parliament rejected the motion, but such large-scale revolt was seen as hurting his authority.
Paradox of UK economy
Despite its anaemic performance, the British economy appears to be profiting from the eurozone crisis, economists have found. As a big economy outside the eurozone, the UK has become a haven for investors, who buy its sovereign debt bonds, shunning other countries such as France.
The interest rate of UK bond yields stood around 2.25%, slightly higher than Germany's (at 1.85%), against 3.3% for France and more than 6% for Italy, Agence France-Presse reported.
Those figures may appear as stunning, in the light of high level of the UK's annual budget shortfall (9.4%), which is higher than Greece's (8.9%). UK's expected growth of 0.7% is among the lowest in the EU and its public debt is comparable to France (84% against 85.4%).
