A former high-ranking member of a European government recently told me about his recurring nightmare. Here’s how he described it: northern Europe turns brown (far right); southern Europe turns red (far left); in the west, Britain falls out of the EU; and in the east, Russia runs over everyone. He could have added: China overtakes the US and runs the world. If you think this scenario is far-fetched, you are in the minority. People almost everywhere are anticipating a major reshuffle of the cards on the global stage. Whether that happens, how and when, will be the defining factors of our age.

These perceptions jump out from a recent Pew Research Centre report examining the state of global opinion about the US, China and the international balance of power. According to the survey, people around the world, especially in Europe, are convinced that China “will eventually replace or already has replaced the US as the leading superpower”. The study doesn’t say whether they believe that is good or bad, nor when they expect these shifts to occur, but it does point to a strongly held expectation.

The survey shows that 59 per cent of Britons and Germans hold the view that China will dominate the globe or already does. Of the French, 66 per cent agree, as well as 60 per cent of Spaniards, 57 per cent of Italians and 46 per cent of Poles. In Asia, opinions are mixed. Of the Chinese, 67 per cent agree their country will be the No 1 superpower or has already risen to that level; 66 per cent of Australians expect China to reach that rank, as do 59 per cent of South Koreans and 53 per cent of Pakistanis. The Japanese disagree, as do the Vietnamese (only 20 per cent and 18 per cent respectively think China will be No 1). In Africa, 47 per cent of those polled said China would overtake the US (33 per cent have the opposite view). In Latin America, it’s 49 per cent (versus 35 per cent).

And in the US, well, opinion is split — 46 per cent of Americans think their country will be overtaken by China or already has been, whereas 48 per cent think the opposite. Whether that says something about national self-doubt in the US, or about hardcore realism, is an open question.

Of course, historical legacies, national ambitions and fears and wishful thinking play a role in this polling. It’s hard to equate the immediacy of a public opinion survey with a more scientific analysis. But public perceptions matter — because, these days, they tend to drive politicians, and not the other way around. So if this is a cautionary tale about the future, then brace yourself for a world in which we will all be on Weibo, and Chinese military bases will face California or Marseille. And prepare to read The Third Woman, the new novel by my Guardian colleague Jonathan Freedland (in which he imagines a world where China rules) as something close to geopolitical prognosis.

Having said that, there are nuances in the Pew report that bear closer scrutiny. Things may not be so clear-cut after all. Take a closer look at the statistics and it appears that two or three years ago more people thought China would dominate the world than they do today. In most of the countries polled, the perception of China’s global clout reached a peak in 2012 or 2013, then dropped several percentage points. The explanation, says Bruce Stokes, Pew director, is that the impact of the great recession of 2008 has gradually worn off. China was seen as coming out as the absolute winner when the West started struggling with bailouts. Now US economic power is perceived to be on the rise again. America’s recovery has bolstered its leadership credentials.

Another interesting pointer from the Pew report is that, on average, global public opinion has a positive image of China. Fifty-five per cent of those polled in 40 countries hold a favourable view of China, against 34 per cent. (In sub-Saharan Africa, the positive views reach a staggering 70 per cent.) But the US receives far higher marks than China in Asia, Latin America and especially Europe. China’s image is severely tarnished by its human rights record. On all continents — with the exception of Africa — majorities say that the Chinese government “does not respect the personal freedoms of its people”, and these majorities are widening. China’s growing economic weight does not translate into support for its political model.

Where does that leave us? As the US and China met this week for a strategic and economic dialogue, speculation continued about whether these two competing behemoths of the 21st century will cooperate on a new global architecture. The post-1945 international order was largely underpinned by US military and economic might. Europe certainly benefited from this. China has bided its time and developed its economy at record speed in recent decades. Now, according to its leader, Xi Jinping, it aims for a “Chinese restoration”. That will breed tensions, just as Germany’s rise in the late 19th century, challenging other European powers, did. Experts say that in the next decade or so, China will become the No 1 economic power, but it still remains far behind the US in military terms. This is a long game. We Europeans now watch mostly from the sidelines. It’s too early to say whether we should be having nightmares or not.

— The Guardian News & Media Ltd