He said the President planned to look at "all" aspects of how the US deals with its historic enemy, which lies just 90 miles south of Florida.
Human rights would be a key part of Mr Trump's revised policy towards Cuba, the press secretary said.
During the final years of the Obama administration, the US softened its stance towards its communist neighbour.
Obama in Cuba
Obama in Cuba
U.S. President Barack Obama makes a face towards a group of children in the audience as he stand on stage with first lady Michelle Obama as he is introduced by Ambassador Jeffrey DeLaurentis Charge d'Affaires to the U.S. Embassy in Cuba
President Barack Obama shakes hands with Cuba's Foreign Minister Bruno Rodriguez upon arrival to the airport in Havana, Cuba
U.S. President Barack Obama and his wife Michelle approach Cuba's foreign minister Bruno Rodriguez as they arrive at Havana's international airport
US President Barack Obama waves after his arrival on Jose Marti Airport in Havana, Cuba
US President Barack Obama carries an umbrella as he is followed by First Lady Michelle Obama after their arrival on Jose Marti Airport in Havana, Cuba
The 'Air Force One' jet carrying US President Barack Obama touches down at the Jose Marti Airport in Havana, Cuba
Mr Spicer's comments suggest Washington may take a harder line with Cuba, raising the possibility of new trade embargoes and the cancellation of commercial flights from the US, which were only recently introduced.
Under Barack Obama, the US ended a policy that allowed any Cuban who made it to American soil to stay and become a legal resident.
Ending the so-called ‘wet foot, dry foot’ rule was welcomed by Havana as “an important step in advancing relations”.
Mr Obama and Cuban President Raul Castro established full diplomatic ties and opened embassies in their capitals in 2015. The former US President visited Havana last March.
But Mr Trump has threatened to end the period of improving relations.
In November he tweeted: “If Cuba is unwilling to make a better deal for the Cuban people, the Cuban/American people and the U.S. as a whole, I will terminate deal.”Reuse content