BRUSSELS — The European Union needs an ambitious grand bargain at its next summit meeting to rescue itself from an accumulation of crises that threaten to blow apart its model of integration.

Like children at a birthday party, each leader has to get a going-home present. And as with many children’s parties, there may be a tantrum along the way.

Chancellor Angela Merkel of Germany is at risk at home from a backlash against a mass influx of Syrian refugees. Prime Minister David Cameron of Britain is trying to win a referendum on staying in the European Union and cannot help her because of public hostility to immigration. Nor can President François Hollande of France, who is struggling for re-election in a country transfixed by the threat from Islamist militants.

Berlin, the Union’s pivotal power, sees controlling migration as the central priority as it tries to cope with one million asylum seekers who have arrived in the last year.