When Angela Merkel arrived at a Stuttgart election rally on Tuesday evening, she had hoped to have in her handbag a package of measures to ease the migration crisis – in Europe and in Germany.

But when Monday's late-night Brussels meeting ended without agreement, the exhausted Dr Merkel arrived at the rally of her Christian Democratic Union (CDU) with little more than a joke.

"I can reveal to you that I nodded off on the flight from Berlin to Stuttgart," she said.

Dr Merkel was in Stuttgart ahead of the “Super Sunday” elections, when one in five Germans go to the polls to choose new governments in three of the country’s 16 federal states.

Normally German state elections are of limited relevance to national sentiment, with results coloured by local issues such as schools and roads. But nothing is normal here since the migration crisis left German television screens and entered daily life, a challenge federal finance minister Wolfgang Schäuble has dubbed Germany’s “rendezvous with globalisation”.

And it is regional politicians on the front lines, requisitioning sports halls to house one million people fleeing far-away wars.

With the next federal election not due until 2017, Sunday’s polls are the first vote of confidence in Dr Merkel’s open-door migration policy and last August’s “we can do this” mantra.

Since then the Merkel mantra has attracted a question mark of doubt, particularly after the New Year’s Eve attacks on women in Cologne, with migrant men among the perpetrators.

Nowhere is the CDU's migration dilemma, and growing doubt in its chancellor, more obvious than in Baden-Württemberg. This prosperous and conservative state in Germany's southwest is home to the precision engineering of Daimler, Bosch and Porsche, and, for more than half a century, was ruled by the CDU.

But creeping complacency saw it ousted in 2011 by the eco-conservativism on offer from local Green Party leader Winfried Kretschmann. After five years Mr Kretschmann is the man to beat in Stuttgart: he enjoys 71 per cent popularity, while his Greens are on 33 per cent and rising.

Local CDU head Guido Wolf, by comparison, has had a miserable campaign. Where the CDU here once polled 40 per cent, it is now five points behind the Greens and falling. And while the Greens have praised Dr Merkel's migration stance as moral and responsible, local CDU officials struggle to hold their tongues.

When Dr Merkel said that EU talks with Turkey to share migrants represented a "breakthrough", Mr Wolf dismissed that assessment before her arrival in Stuttgart as "a little too lofty".

“If people have doubts that the EU [migration] solution will work, I want us to have answers,” said Mr Wolf . “I don’t want people to think about drifting to the right.”

With few answers from Dr Merkel, the local CDU says voters are drifting to the Greens and the Alternative für Deutschland (AfD).

Less than three years after starting life as an anti-bailout party, the AfD has enjoyed a run of success in eastern states as a populist, anti-migrant party. Now it is polling 13 per cent in Baden-Württemberg, neck-and-neck with the centre-left Social Democrats (SPD).

CDU supporters in Stuttgart, squeezed between the Greens and the AfD, gave their leader a reserved welcome in Stuttgart.

“People here are concerned when so many people come from another culture,” Dr Merkel admitted, vowing to deliver succesful integration and a zero-tolerance policy of criminal foreigners.

But people here want Germany’s borders closed. Such national measures are more tempting than Brussels compromises which take longer and “don’t always look elegant”, the CDU leader says.

"But if a well-off continent like Europe says 'this is your problem'," to Turkey, Jordan and Lebanon, she warned, "then this won't go down well in history."

But that is a leap too far for many locals. A decade ago their CDU denied that Germany was a country of immigration. Now the CDU leader wants them to shoulder a massive migration wave in their communities but cannot say who exactly is coming or how they can be integrated.

A similar doubt lingers over the party’s campaign in neighbouring Rhineland-Palatinate. Opposition CDU leader Julia Klöckner, an ambitious 43-year-old, had a good chance at taking office here. Facing voter unease on migration, she lost her nerve and presented demands, including daily asylum quotas. That went down badly and, CDU officials say, damaged her chances of eventually joining the Merkel succession stakes.

Ahead of Sunday, Angela Merkel has warned doubtful CDU regional leaders that she will carry on regardless. They cannot win state elections with Merkel migration policies but, in the long-term, they are lost without her. Asked in Stuttgart if she ever thinks of giving up, she said: “There are many who would wish that, but many who don’t believe in it either.”