BEIJING — Chinese leaders have long sought to present themselves as equals to American presidents. Xi Jinping has wanted something more: a special relationship that sets China apart, as the other great power in an emerging bipolar world.

The Obama administration declined to play along, worried that it implied an American retreat from Asia. But Mr. Xi, the most powerful Chinese leader in decades, may find a more willing partner in President Trump, who is traveling to Beijing this week after stops in Japan and South Korea.

Mr. Trump has often cast China as an unfair trade rival, and, after arriving in Japan on Sunday, he vowed to build a “free and open Indo-Pacific,” a phrase intended to emphasize America’s democratic allies in the region as a balance against China’s rise.

But Mr. Trump has also spoken of China in almost reverential terms and elevated Beijing as a critical player to resolving the North Korean nuclear standoff. And there are signs of mutual admiration between the two leaders — one a Communist Party princeling, the other a brash wheeler-dealer — both of whom see themselves as destined to restore their nations to greatness.