In Friday's travel warning, the State Department said some of the unexplained physical effects have occurred in Cuban hotels, and that while American tourists aren't known to have been hurt they could be exposed if they travel to Cuba.
Rubio is one of five senators on the Intelligence Committee who two weeks ago called on the Trump administration to expel Cuban diplomats from Washington and threaten to shutter the US embassy in Havana if the Cuban government is not more forthcoming about the mysterious attacks that have seriously harmed American diplomats.
On Friday, approximately 60 percent of non-essential staff at the USA embassy in Havana were ordered to leave.
The officials acknowledged Cuba's efforts to investigate the attacks, but stressed that under the Vienna Convention, the government has an obligation to protect foreign diplomats on its territory.
"The Cuban government has never perpetrated nor will it ever perpetrate attacks of any kind against diplomats", Cuba said in a statement on Thursday, when rumors of the embassy staff cut began swirling. Some anonymous officials have alleged that the injuries were due to a "sonic device", potentially planted by Havana or a third-party government such as Russian Federation.
USA authorities first suspected a supersonic device as the cause of the injuries, but investigators have not found any concrete evidence of such.
The AP said other symptoms have been "brain swelling, dizziness, nausea, severe headaches, balance problems and tinnitus, or prolonged ringing in the ears".
The embassy will conduct only core diplomatic and consular functions, including assisting US citizens in Cuba, the official said. FBI investigators and other agents searched and found no devices at the homes and hotels of USA diplomats.
"Due to the travel warning issued by the U.S. Department of State for travel to Cuba, JetBlue will waive change/cancel fees for customers with reservations booked on or before September 29, 2017", the airline announced.
State Department spokeswoman Heather Nauert said that the conversation between Tillerson and Rodriguez Parilla was "firm and frank and reflected the United States' profound concern for the safety and security of its diplomatic personnel". As a result, USA officials are fleeing the country, and the State Department is suspending the processing of us visa applications in Cuba.
However, current United States president Donald Trump, who has announced a partial roll-back of his predecessor's policies, said he would shut the embassy again if necessary.
Tourists drive in a vintage auto past the U.S. Embassy while on a sightseeing tour through Havana on December 18, 2015.
The left rejoiced when President Barack Obama chose to normalize relations with the Cuban regime, reopened our embassy, and allowed travel to the island. Further, the US will stop issuing visas within the country, "effective immediately".
The US will stop sending official delegations to Cuba or conduct bilateral meetings in Cuba for the time being, the senior US official said. So far, Washington is not blaming the Cuban government for the attacks but has made it clear it's unhappy about how Havana is handling the case.
Marco Rubio, a Florida Republican and frequent critic of the Cuban government, called for harsher measures.