Many of the island nations in the region depend to one degree or another on cheap oil imports from Venezuela, a fact that the Caracas government has used as a diplomatic bargaining chip.
So Tillerson stopped off in Jamaica on his way home from South America for talks with senior Jamaican officials, including Prime Minister Andrew Holness, on how to manage any crisis.
On the plane flying in to Kingston, Tillerson told reporters that he had agreed with his Mexican and Canadian counterparts to set up a "very small, very focused working group" on oil sanctions.
A US embargo on Venezuela's main export could deal a decisive blow to President Nicolas Maduro's beleaguered regime, but would also hurt US oil companies operating Gulf Coast refineries.
And it would also -- to the concern of Venezuela's southern neighbors -- hurt ordinary Venezuelans and could be a severe blow to the island economies of the Caribbean, which rely on cheap fuel.
Asked whether US President Donald Trump would green-light an oil ban, Tillerson said: "Well, I don't want to say it's a sure thing because I want to do the work."
But he said he had had productive meetings on Venezuela with the heads of state of Mexico, Argentina, Peru and Colombia.
Soften the impact
In Kingston, Tillerson said he had come to visit Holness because of Jamaica's leading role as the main US ally in the region and the incoming chairman of the CARICOM group.
Both men told reporters the issue of Venezuelan oil had come up, and Holness said that Jamaica was a supporter of democracy no matter what, and could weather an embargo with US help.
"We, now, hardly import oil from Venezuela," he said.
"With the new dynamics in global trade and energy, and the United States is becoming a net exporter of energy resources, Jamaica can in this new paradigm benefit from that."
Tillerson, a former chief executive of US energy giant ExxonMobil, said that he had also discussed with the Jamaicans how their neighbors, some of whom need more oil, would react.
And he suggested that, while no decisions have been made, the United States would be able to put together means to make up for any shortfall and increase Caribbean energy sufficiency.
"I don't want to get into specifics because we're going to undertake a very quick study to see are there some things that the US could easily do with our rich energy endowment, with the infrastructure that we already have available what could we do to perhaps soften any impact of that," Tillerson said.
"There's great unanimity in the region and certainly in the hemisphere that we all want to see some progress on this situation in Venezuela which only gets worse day by day."
Venezuela's Constituent Assembly, a group controlled by Maduro's government, has called an election for April that could see the president returned despite opposition protests and economic turmoil.
Other powers in the region, speaking under Peruvian leadership in the Lima Group, have called on the government to negotiate with the opposition for a return to constitutional democracy.
But Tillerson and the allies he has met in the past five days have agreed that Maduro has not reached out to his rivals and have vowed to reject the "illegitimate" poll.
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