Opposition leader and self-declared interim president Juan Guaido vowed to bring aid in from various points Saturday "one way or another" despite military efforts to block it.
But commanders doubled down on their allegiance to President Nicolas Maduro after Trump warned them to abandon him.
"The armed forces will remain deployed and on alert along the borders... to avoid any violations of territorial integrity," said Defense Minister Vladimir Padrino.
Regional commander Vladimir Quintero later confirmed media reports that Venezuela had ordered the suspension of air and sea links with Curacao and the neighboring Netherlands Antilles islands of Aruba and Bonaire.
Shipments of food and medicine for Venezuelans suffering in the country's economic crisis have become a focus of the power struggle between Maduro and Guaido.
Aid is being stored in Colombia near the Venezuelan border and Guaido aims also to bring in consignments via Brazil and Curacao.
A Brazilian presidential spokesman said the country was cooperating with the United States to supply aid to Venezuela but would leave it to Venezuelans to take the goods over the border.
Maduro says the aid plan is a smokescreen for a US invasion. He blames US sanctions and "economic war" for Venezuela's crisis.--AFP