In August, Washington doubled the bounty on Mr Maduro’s head, offering $50m for information leading to his arrest for alleged drug trafficking.
The build-up of American troops and military hardware in the region in recent weeks has sent a clear signal to the president and his key lieutenants that they are living on borrowed time.
The New York Times reported on Tuesday that Mr Maduro had become increasingly reliant on Cuban bodyguards and sleeps at a different location each night.
Amid the escalating tensions, the Trump administration has steered away from talking about regime change.
However, in their phone call, the American president offered Mr Maduro a soft option version of a soft way out.
The leader would be given amnesty with his wife and son and a chance to flee the country if he promised to resign on the spot.
‘A rock and a hard place’
It is understood that some of the discussion centred on Mr Maduro’s desire to live a comfortable life in exile.
He asked to leave with $200m (£150), two sources confirmed, possibly from funds already frozen by US authorities.
That was not the sticking point.
More problematic was his demand that amnesty be given to dozens of his associates and that key allies would run a transitional government that would eventually oversee free elections.
One of the sources familiar with the call said Mr Maduro “was caught between a rock and a hard place”.
He could face reprisals from senior figures if he agreed to a deal that left close allies to face justice.
“It’s self-preservation, not altruism,” said the source.
But that demand was unacceptable to US officials. They have made it their mission to bring down the Cartel de los Soles, a network of military and political officials who run much of the country’s drug trade.
A broad amnesty could leave much of that cartel, designated by Washington as a “foreign terrorist organisation”, in place.
A victory for Trump?
Vanessa Neumann, who served as envoy for the Venezuelan opposition to London, said it was no surprise that both sides were trying to find an off-ramp for Mr Maduro.
“The thing that Trump has to weigh would be whether Maduro exiting would be viewed as a win for him,” she said.
“Everybody plays to their own local constituency. So if you have a big part of Cartel de Los Soles remain intact, how much of a victory is that for Trump?”
The White House faces a ticking clock as it weighs its next move.
Elements of Mr Trump’s Maga base are becoming restless at the thought of a new foreign adventure.
Maintaining the huge military presence in the region is costing hundreds of millions of dollars. The price of keeping the USS Gerald R Ford carrier strike group on station in the Caribbean is estimated at more than $6m every day.
