Suleiman also served as Syria's liaison officer to Lebanon's Hizbullah movement.
According to the report, Suleiman was shot dead by a sniper in the Syrian port city of Tartous. The assassin's identity is unknown.
It was also reported that the officer's death was not reported in Damascus, and that it appears the Syrian authorities are exerting efforts not to bring the affair to the public's attention.
The London-based Arabic-language newspaper al-Hayat reported that a senior army officer close to Assad was killed Friday night. The report did not name the officer, but said he was responsible for a series of sensitive matters.
About half a year ago, the Lebanese organization's operations officer Imad Mugniyah, another key figure operating on the Syria-Hizbullah axis, was assassinated in Damascus a mysterious blast in his car.
Media outlets across the world linked Israel to the assassination, although Mugniyah had several "unsettled accounts" with quite a few elements worldwide.
Over the weekend, the man responsible for southern Lebanon on behalf of Hizbullah warned that his organization was "stronger than ever and ready to confront Israel." Nabil Kaouk spoke of the relations between Hizbullah and Iran, stressing that the option for a war against Israel remains opens.
Defense Minister Ehud Barak warned during a meeting with UN Secretary-General Ban Ki-Moon in New York last week that "Hizbullah's armament may undermine the delicate balance in Lebanon." He added that "Israel views this as a real and serious danger."