He and President Donald Trump reviewed the options in a White House meeting this week, officials said.
The State Department also issued a travel warning - telling US citizens to avoid travel to Cuba. He also accused the US of politicizing the incidents.
The U.S. government has ordered 60 percent of its staff removed from the diplomatic headquarters in Havana.
The U.S.is removing all its its "nonessential" embassy staff, according to the report, and will stop processing visas "indefinitely" in response to what sources told the AP were "specific attacks" on its staff.
On the same day of the Tillerson-Rodriguez meeting, sources familiar with the investigation said that the Trump administration does not believe Cuba is responsible for the attacks against USA embassy staff.
The attacks occurred in residences and hotels frequently used by Americans.
At a meeting in Washington this week Tillerson reminded Bruno Rodriguez, Cuban foreign minister, of the country's obligation under the Vienna Convention to protect diplomats and their families.
The US expelled two Cuban diplomats in late May over the incidents.
"Protecting the safety of our diplomats is of utmost importance". The American Foreign Service Association reported in early September that symptoms include mild traumatic brain injury, permanent hearing loss, loss of balance, headaches and brain swelling.
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Tillerson, for his part, told the visiting Cuban that the USA still has "profound concern" for the safety and security of its diplomats in Havana.
Between five and 10 Canadian diplomats and their families suffered sonic attacks similar to the Americans. Vague "sonic attacks" were initially blamed for the symptoms, which ranged from concussions and hearing loss to nausea and dizziness. Cuba said it was investigating the incidents. Clues about the circumstances of the incidents make any explanation scientifically implausible.
Secretary of State Rex Tillerson said he had chose to trim staff and warn Americans in response to what he called "attacks of an unknown nature".
The relations between the United States and Cuba will tend a little more this Friday. Cuba described his comments as "unacceptable and meddling". The embassy, an iconic building on Havana's waterfront, also will suspend processing USA visa applications. Canadian diplomats in Cuba also reported similar injuries in March 2017.
The case has brought simmering tensions between the two countries since Trump took office to the boil.
Although the USA has opted against fully closing its embassy (which was reopened in 2015 as part of President Barack Obama's reestablishment of diplomatic relations with Cuba after more than half a century of tensions), the steps represent a significant blow to diplomatic ties. And the Trump administration has stopped short of blaming Cuba for the attacks, leaving open the possibility that another country or group is responsible.
The attack may have been the work of a country like Russian Federation or a rogue government unit, The New York Times reported.
How have USA officials and leaders reacted? Staffing at the embassy in Havana was already lower than usual due to recent hurricanes that whipped through Cuba. The top U.S. diplomat has called them "health attacks".
United States officials say there may have been as many as 50 attacks, a senior United States official told CNN, the most recent in August.
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