Clamping down on engagement with Cuba would be a high-profile way for Mr. Trump to showcase a stark break with his predecessor and to fulfill a pledge, delivered during a speech in Miami in September, to a crucial constituency that disproportionately supported him. It would also enable the president to reward the loyalty of Cuban-American lawmakers who have been agitating for a harder line on Cuba, including Senator Marco Rubio and Representative Mario Diaz-Balart, both Republicans of Florida.

But as the White House has sought to formulate a series of steps for Mr. Trump to announce, a split has emerged over rolling back a policy that many senior officials privately agree has been an improvement on the Cold War dynamic that shaped relations with Cuba in the past. In addition to the revival of diplomatic relations for the first time in a half-century and liberalized rules for trade, travel and commerce, the new approach has paved the way for cooperation in intelligence-sharing, drug interdiction, scientific research and a host of other areas.

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"I am confident the president will keep his commitment on Cuba policy by making changes that are targeted and strategic and which advance the Cuban people's aspirations for economic and political liberty," said Mr. Rubio, who has met with and talked to Mr. Trump and his top aides several times on the matter.

As the White House labored in March to corral Republican votes for an unpopular health care overhaul measure, Mr. Diaz-Balart asked for assurances from Mr. Trump that he would hold to the hard line on Cuba he laid out in his campaign. The Florida Republican supported the measure and has played an influential role in shaping the new Cuba policy.