The Yisrael Hayom daily said Peres' body guards were on heightened alert after tips the president is being targeted, Arutz Sheva reported.
Other senior officials could also be targeted with the approach of the anniversary of the death of Imad Mughniyeh, killed Feb. 12, 2008, by a Damascus car bomb.
Hezbollah blamed Israel for his death.
Security was increased at events Peres has attended in and around a rented apartment in Jerusalem where he is staying after flooding of the Presidential Residence.
A confrontation between Peres' bodyguards and three Arabs from eastern Jerusalem took place outside the president's apartment Sunday when the Arabs cursed and made threats, officials said.
One of them was arrested and two escaped, the officials said.
Peres was not in the apartment at the time.