Mr Tillerson raised the prospect of a Venezuelan military coup ahead of a five-day tour of Latin America.
Discussing Venezuela, the US official said militaries in Latin America often “handled” transitions from bad governments, but insisted he was not advocating “regime change.”
In a speech at the University of Texas in Austin he said: “In the history of Venezuela and South American countries, it is often times that the military is the agent of change when things are so bad and the leadership can no longer serve the people.
“If the kitchen gets a little too hot for him, I am sure that he’s got some friends over in Cuba that could give him a nice hacienda on the beach and he could have a nice life over there.”
Mr Tillerson added that then when the people’s patience begins to fade Venezuela’s military “will manage a peaceful transition”.
The Secretary of State avoided using words such as “military coup” or “overthrow”, and insisted Washington is not plotting to help oust Mr Maduro, who has a close relationship with Cuba’s communist government.
He said: “We have not advocated for regime change or removal of President Maduro.
“We have advocated that they return to the constitution.”
Venezuela condemned Mr Tillerson’s comments and claimed the US Secretary of State was seeking a regional “intervention” against the socialist government.
Accusing Washington of seeking to undermine democracy in Latin America and return to the days of “imperialism,” Defense Minister Vladimir Padrino lashed out at Mr Tillerson during a broadcast on state television.
Mr Padroni said: “Every day he distances himself more from diplomacy to enter into war-mongering. You have no moral authority.
“This man will try to persuade Latin America governments to intervene in Venezuela. That’s a publicity stunt.”
Many opposition leaders would love for Mr Maduro to seek exile abroad. The former bus driver and union leader is widely disliked in Venezuela due to a brutal economic crisis that has sparked malnutrition, disease and mass emigration.
Washington did not recognise the latest national election in Venezuela, which led to the inauguration of the National Constituent Assembly in August.
The US responded by slapping Venezuela with sanctions targeting the financial and energy sectors of the oil rich nation.
The Constituent Assembly, drawn up to rewrite the constitution, took over the opposition-led National Congress.
The election campaign and its aftermath were marred by violence and opposition protests. The largest pro-government demonstrations took place in the capital Caracas.
The US has slammed Mr Maduro for running what is called an “authoritarian dictatorship”.
Mr Maduro has accused the USA of “aggression” and meddling in Venezuela’s international affairs in order to bring about regime change.
He went so far as to say the army should be prepared to defend the country as it was being “blatantly threatened by the most criminal empire in the history of mankind.”
US President Donald Trump has said in the past: “A military option is certainly something we could pursue”.
The Pentagon has denied it has been tasked with devising a potential military scenario of Venezuela.
The country’s socialist party has picked Mr Maduro as their candidate for this year’s presidential vote.