"To all Airlines, Pilots, Drug Dealers, and Human Traffickers, please consider THE AIRSPACE ABOVE AND SURROUNDING VENEZUELA TO BE CLOSED IN ITS ENTIRETY," Trump said in a Truth Social post.
US officials contacted by Reuters were surprised by Trump's announcement and unaware of any ongoing US military operations to enforce a closure of Venezuelan airspace.
The Pentagon did not respond to requests for comment and the White House did not provide any further explanation.
Venezuela's communications ministry, which handles all press inquiries for the government, did not immediately reply to a request for comment on Trump's post.
David Deptula, a retired lieutenant general who commanded a no-fly zone over northern Iraq in 1998 and 1999, said Trump's announcement raises more questions than it answers.
Imposing a no-fly zone over Venezuela could require significant resources and planning, depending on the goals of the airspace closure, he said.
"The devil's in the details," Deptula said.
Trump's administration has been weighing options to combat what it has portrayed as Venezuelan President Nicolas Maduro's role in supplying illegal drugs that have killed US citizens.
The socialist Venezuelan president has denied having any links to the illegal drug trade.
The US military is poised for a new phase of operations after a massive military build-up in the Caribbean and nearly three months of strikes on suspected drug boats off Venezuela's coast.
Maduro, in power since 2013, has contended that Trump is seeking to oust him and that Venezuelan citizens and the military will resist any such attempt.
The streets of Caracas were largely quiet on Saturday morning, although some people braved rain to go shopping.
Trump's announcement on Venezuela's airspace followed a warning last week from the US Federal Aviation Administration that major airlines faced a "potentially hazardous situation" when flying over Venezuela due to a "worsening security situation and heightened military activity in or around" the country.
Venezuela revoked operating rights for six major international airlines that had suspended flights to the country after the FAA warning.
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