Meanwhile, Senator Marco Rubio, another Republican and a fierce critic of Cuban President Raul Castro, has thrown his weight behind Trump. A group of 55 Cuban women entrepreneurs sent a letter to Ivanka Trump to let her know the change in policy is going to hurt them.
One of the worst-kept secrets in official Washington this week is that the president will jet to Miami Friday and formally announce the reversal of President Obama's 2015 orders that opened Cuba up to the US after more than half a century.
Havana's national statistics bureau said that by the end of May, some 284,565 Americans had travelled to Cuba this year.
Tillerson's remarks come as the Trump administration is conducting an interagency review of its Cuba policy, which is expected to result in a roll back of some Obama-era changes.
Theirs is a distinctly minority view - in Congress, in the business community and in the general population, even among Miami-Dade County's Cuban-Americans.
"Thousands of Americans are visiting Cuba and fueling the fastest growth in its private sector since 1959", CubaOne, a group of young pro-engagement Cuban-Americans, wrote in an open letter to Trump Monday.
"If Cuba is unwilling to make a better deal for the Cuban people, the Cuban/American people and the USA as a whole, I will terminate deal", Trump said in a tweet.
Although many Americans support Obama's decision and the United States business community has welcomed moves to reopen trade, Trump's hardline campaign rhetoric won him support among influential Cuban exiles in Florida.
Critics of Obama's approach contend that many US visitors have taken advantage of eased regulations and looser scrutiny to visit the island for pleasure trips. Existing restrictions on business dealings with the Cuban government, and especially its economically powerful military, will be made more legally explicit, and at least some planned expansions of commercial activity will be frozen. Rep. Mario Diaz-Balart openly acknowledges that he extracted assurances from Trump that he would get tougher on Cuba before he, the congressman, would agree to support the president's unpopular health care overhaul.
That fine has been the first of its kind imposed by the White House since Trump took office.
A White House spokeswoman on Tuesday said only: "We do not have any declarations regarding Cuba policy at this time, nor do we have a confirmed time or date for an announcement". A recent report by the Boston Consulting Group said USA visitors to Cuba could rise by as much as sevenfold by 2025.
Meanwhile, Sen. Chris Murphy joined a group of senators Monday asking Secretary of State Rex Tillerson, Murphy to build on progress in Cuba for the benefit of USA workers and businesses competing for opportunities. An educated generation now rises to power yearning for more.
But the measures have proven popular with the public, US businesses and many lawmakers from both parties. The number of US visitors who are non-Cuban American increased by 74 percent in 2016, facilitated by the restoration of commercial flights between the two countries. American visits would likely drop, hurting both USA airlines and accommodation services operating there, as well as the Cuban tourism industry.
Many Cubans, like Coba, had been banking on the number of American arrivals continuing to grow.