According to the newspaper, he raised this issue during his visit to Moscow during meetings with representatives of the Federal Security Service (FSB) and the Foreign Intelligence Service (SVR) of Russia.
The CIA declined to comment on Burns' warning.
Earlier, the director of the Russian Foreign Intelligence Service, Sergei Naryshkin, in an interview with RT commented on reports of the Havana syndrome, which American diplomats complained about, including from the US Embassy in Cuba.
So-called acoustic attacks have been reported by US diplomats in Cuba and China since 2016.
Embassy officials complained of unusual symptoms, including high blood pressure and dizziness.
The US National Academy of Sciences, Engineering and Medicine suggested that it could be about the impact of "directed impulsive radio frequency energy", but also chemical exposure, infectious diseases and psychological problems are not excluded.