This hole is largely of his own making. In 2013, he buckled under pressure from his party’s euroskeptic wing and agreed to hold a referendum on Britain’s continued membership of the European Union. This was a significant defeat for a prime minister who had previously pleaded with his party to “stop banging on about Europe.” To soften the blow, Mr. Cameron came up with a ruse: he committed only to holding the referendum by the end of 2017, buying him time, so he said, to renegotiate the terms of Britain’s EU membership.

This gambit succeeded in the short term since it secured a truce within the Conservative party sufficient to get it past the election. But it also planted a bomb that now threatens to blow up his second term: Having insisted that Britain’s relationship with the EU is broken, he now needs to fix it before he can credibly lead the campaign for Britain’s continued membership.

This would not be such a problem if there were, in fact, something obviously wrong with the terms of Britain’s membership of the EU. But the government’s own comprehensive review last year failed to identify any serious flaws. Mr. Cameron has talked vaguely of exempting Britain from the EU Treaty’s commitment to “ever closer union of the peoples,” of making the EU more competitive, more democratic, less bureaucratic, and safeguarding the U.K.’s financial-services industry from an integrating eurozone. But this hasn’t yet translated into any list of specific demands, let alone identified anything so vital to the U.K. national interest that left unfixed would justify quitting the EU.

Instead, Mr. Cameron and his officials have been fanning out across Europe in a frantic search for something—anything—that is both deliverable in the EU and can provide the political cover that he and many of his colleagues believe they need to throw themselves into the campaign to keep Britain in the EU. This is proving increasingly difficult. Mr. Cameron has already quietly shelved plans to secure British opt-outs from EU social and employment rules and human-rights law. Meanwhile, Mr. Cameron’s single specific demand—to restrict in-work welfare payments to EU migrants—has run into a brick wall of resistance from EU governments unwilling to sanction discrimination against their citizens.

Euroskeptics are convinced that Mr. Cameron’s renegotiation is going nowhere—or at the very least that it won’t secure the blanket opt-outs and future vetoes over almost everything that the EU does that they are demanding. As a result, some are starting to break ranks with Mr. Cameron and campaign openly for an EU exit. That adds to the pressure on Mr. Cameron: The more noisily euroskeptics campaign, the more pressure the government feels to talk up the significance of its renegotiation and to make more far-reaching demands.

What Mr. Cameron won’t do is to give his full-throated backing to Britain’s EU membership. Certainly, he will attempt to say as little as possible on the subject at his party conference to avoid further inflaming euroskeptic passions, says a senior party official. The government still believes that it can achieve something of strategic significance from its negotiations and that to speak too positively about Europe now would undermine its negotiating position at a time when it wants to keep open the possibility that it might campaign to leave.

Yet even some of those closest to Mr. Cameron worry he is leaving it very late to make the case for EU membership. “You have to fatten the pig before you take it to market,” says one senior party official. Mr. Cameron has barely acknowledged the EU Commission’s substantial efforts to boost competitiveness, cut red tape, deepen the single market and secure new free-trade deals, including with the U.S.—all key U.K. objectives. Similarly, Mr. Cameron eschewed the opportunity provided by the current migration crisis to explain the central role that the EU can and must play in forging a common European response to common European challenges.

Mr. Cameron’s silence is in turn creating a dilemma for other pro-Europeans. Business and finance leaders have come under pressure from Downing Street to get behind the prime minister’s renegotiation gambit, talking up the need for reforms rather than making the case for EU membership. They have been told that the best chance of keeping Britain in the EU is if all pro-Europeans work together to help Mr. Cameron out of his hole.

Yet by failing to challenge the euroskeptics, they may, in fact, increase the risk that Mr. Cameron’s gambit will fail—and that he ends up dragging them into his hole too.