BEIJING — An anonymous letter calling on President Xi Jinping to resign for the good of China and his own safety seemed to be digital rumor-mongering when it appeared on the Internet this month. It spread by email and lingered on a small domestic Chinese news site before it was removed.

But the response from Beijing has been anything but dismissive.

Surprising even some hardened critics, Mr. Xi’s security forces have overseen a far-reaching inquisition to root out the culprits behind the letter, resorting to measures that have drawn more attention than the letter itself. They have detained at least 11 people, including relatives in China of two exiled writers accused of spreading or promoting the letter.

Mr. Xi’s handlers have sought to give him an aura of unshakable dominance. But the unusually severe response to what might be nothing more than an outlandish Internet ruse suggests some anxiety about his hold on power, including among security officials keen to show their loyalty and avoid any hint of exposing him to danger, experts said.

“The response has shown how jittery they are,” said Kerry Brown, professor of Chinese politics at King’s College, London. “The fear seems to be that these views might be taken as representative of real elite figures.”