The Phileleftheros newspaper said the 26-year-old man, who was remanded in custody on Thursday, belonged to the Lebanese organisation’s military wing and had personal links to Hezbollah leader Hassan Nasrallah.
The daily said the man, who has not been named, arrived in Cyprus last week for what he described as a holiday.
He was arrested on Wednesday following a period of surveillance and more than 400 boxes of ammonium nitrate — a fertiliser that when mixed with other substances can be used to make explosives — was discovered at his home in Larnaca.
The suspect reportedly told police he knew the material was in his basement but claimed it was not his.
The Politis daily said authorities had not ruled out that the suspect was planning an attack on Israeli targets on the Mediterranean resort island, which attracts thousands of visitors from nearby Israel each year.
“We are investigating every possibility and any links (to extremism), if they exist, will be investigated and evaluated,” said Justice Minister Ionas Nicolaou.
Police sources say the man faces possible charges of conspiracy to commit a crime, membership of a terrorist organisation and illegal possession and transfer of explosive materials.