When Facebook CEO Mark Zuckerberg delivered a speech about freedom of expression in Washington, D.C., last fall, there was also another agenda: to raise the alarm about the threat from Chinese tech companies and, more specifically, the popular video-sharing app TikTok.

Tucked into the speech was a line pointing to Facebook’s rising rival: Mr. Zuckerberg told Georgetown students that TikTok doesn’t share Facebook’s commitment to freedom of expression, and represents a risk to American values and technological supremacy.

That was a message Mr. Zuckerberg hammered behind the scenes in meetings with officials and lawmakers during the October trip and a separate visit to Washington weeks earlier, according to people familiar with the matter.