The United States has deployed several Navy ships and thousands of troops near Venezuelan waters. The Trump administration says the military buildup is intended to target Venezuelan drug shipments to the United States.

This week, the moves took a drastic turn. President Trump said Tuesday that U.S. forces had killed 11 people in a strike on a boat in the southern Caribbean that he said was transporting drugs from Venezuela. He posted a video that he said depicted the attack. Venezuela’s government claimed, without offering evidence, that the video was made with artificial intelligence.

U.S. officials have said that Venezuelan cocaine shipments are contributing to overdose deaths in the United States and that cocaine is often laced with fentanyl. They accuse the country’s autocratic leader, Nicolás Maduro, of overseeing a narcotics cartel.

Mr. Maduro, at a news conference on Monday in the capital, Caracas, called the naval buildup “the greatest threat our continent has seen in the last 100 years.”

What exactly is Venezuela’s role in the drug trade? And does Mr. Maduro have links to the illicit business?