The European Union is facing one of its roughest power struggles over the most influential job in Brussels, as Europe's leaders seek to reassert control after an election that saw big gains for the far right and the hard left, and amounted to a vote of no confidence in many of the national governments.

David Cameron has been pushed on to the defensive after revelations of deep disagreement last week over the candidacy of the Luxembourg federalist Jean-Claude Juncker for the post of European commission president. But he is not the only one taking a stand against the leading candidate.

On Sunday, the Italian prime minister, Matteo Renzi, signalled that Juncker may not be acceptable to him. And after losing the elections in France to the far-right Front National, François Hollande is also said to oppose Juncker, allegedly telling the German chancellor, Angela Merkel, that the job should go to the French, preferably his former finance minister Pierre Moscovici.

Merkel, Europe's most powerful leader, is seen as the pivotal figure in seeking a compromise over the commission post. She has remained equivocal over Juncker's prospects, although formally, as Germany's Christian Democratic leader, she supports him.

Cameron told Merkel at a summit last week that Europe could not be run for the next five years by "a man of the 1980s", according to Der Spiegel.

This is a battle that has been in the offing for months. Before the election, the Guardian reported that Cameron would have a problem with Juncker because he was the head of a parliamentary bloc to which no British party belonged. Juncker told the Guardian then: "Cameron has to stick to the clear treaty rules … Whoever wins, wins … The question is not whether we are supported in Great Britain. The question is rather why does Great Britain not stick to the vote of the continental Europeans?"

Juncker reiterated his frustrations in an interview published in Bild Zeitung on Sunday. "We cannot allow ourselves to be blackmailed," he said.

The arguments over who should get the commission job, as well as a clutch of other top jobs in Brussels, are likely to continue until the autumn. The view in Brussels is that Cameron blundered by moving so quickly to oppose Juncker. He is also seen as weak on Europe because of his pledge of a referendum on Britain's membership in 2017, and because he is perceived as a hostage of his backbenches and of Nigel Farage's Ukip, which trounced the Conservatives in the election, driving the Tories into third place.

Cameron is supported most vocally by Hungary and also by the Swedish and Dutch prime ministers, but he cannot veto Juncker's appointment and so far has not enough support to erect a blocking minority in the European council.

Although national leaders are at odds over who should get the commission post, the real fight is between the governments of the EU and the European parliament over who should have the defining say in making the appointment. At the insistence of the main parties in the parliament, last week's election was the first to include nominees for the commission post, meaning that the candidate of the winning party should automatically get the job.

But it is up to national leaders, not the parliament, to propose a commission president. The nominee then has to be endorsed by an absolute parliamentary majority of 376. Parliament leaders are threatening to block anyone who is not their choice.

"I am now conducting all of my talks in the spirit that Jean-Claude Juncker should become president of the European commission," Merkel said on Friday, in her strongest declaration of support for the Luxembourger.

She left herself wiggle room, however. Her top EU advisers in Berlin are known to oppose Juncker on the basis that the parliament has to be shown who takes the decisions.

The president of the council chairing EU summits, Herman Van Rompuy, is also a vocal opponent of having directly elected candidates for the post. He is in charge of negotiations with parliament leaders aimed at crafting a compromise and striking a deal.