Trump said Venezuela's interim president Delcy Rodríguez could "pay a very big price, probably bigger than Maduro" if she "doesn't do what's right".
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His comments to US magazine The Atlantic came as the country's deposed president Nicolás Maduro was set to appear in a New York court on Monday.
The US accuses Maduro, who is charged with drug trafficking and weapons offences, of running a "narco-terrorist" regime, a claim he denies.
Rodríguez - who served as Maduro's vice president - adopted a conciliatory tone at her first cabinet meeting on Sunday, offering co-operation with the US.
"We invite the US