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When the European Union unexpectedly won the Nobel Peace Prize this month, the leaders of Germany, France and Italy spoke of their pride. But the British prime minister, David Cameron, maintained an awkward silence.

Before that, the British government said it wanted to exercise an opt out of an estimated 133 areas of European Union police and judicial cooperation to which it had once agreed.

And Mr. Cameron supported a plan for a new budget for countries that use the euro (which Britain does not), something that would place his nation firmly in Europe’s outer tier. The prime minister has been hinting that he could hold a referendum on Britain’s relations with the union, and one newspaper reported recently that a senior cabinet minister wants Britain to threaten openly to leave the 27-nation bloc. There was no official denial of the report.

All of which has fueled concerns that Britain is laying plans for what political and financial pundits have dubbed “Brixit,” a variant on “Grexit” — the shorthand for Greece’s much predicted if currently forestalled departure from the euro zone.

Mr. Cameron insists that he is trying to keep Britain in the European Union. He argues gamely that popular consent to membership can be regained only by refocusing the relationship on Europe’s single economic and free trade market — which accounts for half of Britain’s foreign trade and investment, according to the government — and loosening other ties.

Britain has always been ambivalent about the European project. Unlike the founding six nations, all of them defeated or occupied in World War II, Britain was a victor. In national mythology, the war was neither a moment of disgrace nor a humiliation. On the contrary, it was widely considered the country’s finest hour, when it stood alone against fascism.

So the idea of reconciliation through integration never had the appeal in Britain that it did on the Continent. Unlike many other member countries, Britain always paid more into the union in contributions than it received in subsidies.

Now, with the euro zone almost three years in crisis, British public opinion has hardened. The overwhelming majority of Conservative lawmakers are euro skeptics, and many privately favor withdrawal.

For some this is a question of conviction, while others feel a competitive threat from the United Kingdom Independence Party, which wants to take Britain out of the union altogether. Adept at winning over Conservative voters, the party threatens to deprive many Tories of their seats in Parliament in future elections.

So government strategy toward the union — always hampered by what Lord Christopher Patten, a former Conservative minister and ex-European commissioner, has called “the psychodrama of Britain’s relations with Europe” — has turned on its head.

When he was prime minister, Tony Blair sought to exploit strains between France and Germany, the twin engines of European integration. Mr. Blair, whose Labour Party was less Europe-averse than Mr. Cameron’s Conservatives, courted allies among smaller nations and tried to compensate for Britain’s self-exclusion from the euro by leading in areas like defense and police cooperation, a policy Mr. Cameron has reversed.

Previous British governments argued that if they did not like something, they had a chance of changing or stopping it only if they sat at all tables with their European partners.

Mr. Cameron seeks a new arrangement that abandons any pretense of being at the heart of the European Union. He does not, for instance, want to stop the euro zone integrating without Britain. Indeed, he recognizes that this is necessary to save the euro.

But can a more remote relationship work?

According to a recent study for the European Council on Foreign Relations by Peter Kellner, president of YouGov, a polling organization, there is a parallel with 1975, when Britain held its referendum on membership in Europe.

“Then, as now, the prime minister, then Labour’s Harold Wilson, had a problem managing party divisions,” Mr. Kellner wrote. “Then, as now, most voters wanted to leave the Common Market (as it then was). Then, as now, polling (specifically, a Gallup Poll in November 1974) suggested that if the prime minister renegotiated the terms of Britain’s membership and recommended acceptance of the new terms, opinion would swing in favor of British membership.”

Mr. Kellner went on to note that Mr. Wilson did talk to his European partners, claimed victory and voters subsequently voted 2 to 1 to stay in Europe.

In July this year, a YouGov poll suggested that, if Mr. Cameron renegotiated the relationship to his satisfaction and recommended a “yes,” 42 percent of voters would vote to stay in and 34 percent to leave.

The strategy may have domestic political logic but there are simultaneous risks: of reducing Britain’s influence on the world stage and making a “Brixit” a self-fulfilling prophecy.

Britain carries weight with some other member states who rely on British influence to bolster the bloc’s free-market wing and counterbalance France’s more statist approach.

But to anchor Britain in Europe, Mr. Cameron needs to emerge from a whole series of negotiations successfully — or at least persuade his own skeptical party that he has done so.

Most urgently, he faces tough discussions on the European Union’s next seven-year spending cycle. Many officials and other observers expect Mr. Cameron to veto a budget deal at a November summit.

That will satisfy euro skeptics only if Mr. Cameron can bring home an improved offer later. Yet, playing to his domestic gallery with an aggressive veto may alienate the very European allies Mr. Cameron would need in later talks in any effort to redefine ties.

Meanwhile, the emergence of an ever more clear-cut two-tier Europe, with much greater integration among the 17 euro zone nations on issues like banking and financial services, is putting a strain on Europe’s unified economic space, and could ultimately threaten London’s status as Europe’s financial capital.

“Deeper integration in the core would come with disintegration in the E.U.’s periphery and shrink the single market,” writes Sebastian Dullien in a separate paper, also for the European Council on Foreign Relations. In other words, it could undermine the one part of the European bargain that Britons actually seem to like.