At least 21 Americans and five Canadian families have been hurt since last year in what Tillerson has described as "health attacks" and which US officials think may have come from an "acoustic device."
But last week, Cuba's Foreign Minister Bruno Rodriguez Parrilla told the UN General Assembly that his country had found no evidence to link the symptoms to any outside assault.
Tomorrow, the top diplomats were to meet behind closed doors at the State Department to try to resolve the worst crisis in US-Cuba relations since ties were restored in 2015.
"We cannot rule out additional new cases as medical professionals continue to evaluate members of the embassy community. We do not have definitive answers on the source or cause of the incidents."
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Tillerson said last week that the United States is considering whether to close its embassy in Cuba following the alleged attacks, which US authorities say began last year and continued to at least last month.
According to a union representing US overseas staff, the victims have suffered a variety of symptoms and injuries, including in some cases brain trauma and hearing loss.
"Cuba has never perpetrated nor will it ever perpetrate actions of this sort. Cuba has never allowed nor will it ever allow its territory to be used by third parties with that purpose," Parrilla said.
Relations between the United States and Cuba were restored by then president Barack Obama and his counterpart Raul Castro in 2015, half a century after the Cold War rivals cut ties and began a long diplomatic standoff.