"We're going to run the country until such time as we can do a safe, proper and judicious transition," Trump said during a press conference at his Mar-a-Lago resort in Florida on Saturday.
"We can't take a chance that someone else takes over Venezuela who doesn't have the interests of Venezuelans in mind."
It is unclear how Trump plans to oversee Venezuela.
Despite a dramatic overnight operation that knocked out electricity in part of Caracas and captured Maduro in or near one of his safe houses, US forces have no control over the country itself, and Maduro's government appears to still be in charge.
Trump's comments about an open-ended presence in Venezuela echoed past leadership changes in Iraq and Afghanistan, both of which ended in US withdrawals after years of occupation.
He said he was open to the idea of sending US forces into Venezuela.
"We're not afraid of boots on the ground," he said.
Trump did not provide specific answers to repeated questions from reporters about how the US would run Venezuela.
US occupation "won't cost us a penny" because the United States would be reimbursed from the "money coming out of the ground," Trump said, referring to Venezuela's oil reserves, a subject he returned to repeatedly during Saturday's press conference.
Trump said US Secretary of State Marco Rubio had been in touch with Venezuelan Vice President Delcy Rodriguez - Maduro's presumptive successor.
"'We'll do whatever you need,'" Trump quoted Rodriguez as saying.
"She really doesn't have a choice."
Reuters could not immediately corroborate the exchange.
Four sources familiar with her movements said Rodriguez was in Russia. The Russian foreign ministry dismissed the report about Rodriguez's presence in their country as "fake."
The removal of Maduro, who led Venezuela with a heavy hand for more than 12 years, potentially opens a power vacuum in the Latin American country.
Any serious destabilisation in the nation of 28 million people threatens to hand Trump the type of quagmire that has marked US foreign policy for much of the 21st century, like the interventions in Afghanistan and Iraq.
The US has not made such a direct intervention in its backyard region since the invasion of Panama 37 years ago to depose military leader Manuel Noriega over allegations that he led a drug-running operation.
The United States has leveled similar charges against Maduro, accusing him of running a "narco-state" and rigging the 2024 election.
Maduro, a 63-year-old former bus driver handpicked by the dying Hugo Chavez to succeed him in 2013, has denied those claims and said Washington was intent on taking control of his nation's oil reserves, the largest in the world.
Venezuelan officials condemned Saturday's intervention.
"In the unity of the people we will find the strength to resist and to triumph," Defense Minister Vladimir Padrino said in a video message.
While various Latin American governments oppose Maduro and say he stole the 2024 vote, direct US action revives painful memories of past interventions and is generally strongly opposed by governments and populations in the region.
Trump's action recalls the Monroe Doctrine, laid out in 1823 by President James Monroe, laying US claim to influence in the region, as well as the "gunboat diplomacy" seen under Theodore Roosevelt in the early 1900s.
Venezuelan allies Russia, Cuba and Iran were quick to condemn the strikes as a violation of sovereignty.
Tehran urged the UN Security Council to stop the "unlawful aggression."
Among major Latin American nations, Argentina's President Javier Milei lauded Venezuela's new "freedom" while Mexico condemned the intervention and Brazil's President Luiz Inacio Lula da Silva said it crossed "an unacceptable line."
