An Al-Watan report Monday says the family and Assad aides are residing on the ship in the Mediterranean Sea and that he travels to Syria by helicopter to attend official meetings and receptions.
Otherwise, he stays on the warship, the sources told the Arabic language newspaper.
When he flies to his embattled country, the president lands at undisclosed locations and is transported to the presidential palace under heavy guard, the sources said.
The Russian-guarded warship provides a safe environment for Assad, who has lost confidence in his own security detail, the report said.
Assad's presence on the warship suggests he has been granted political asylum by Russia but there has been no official comment from Moscow, the newspaper said.
The circumstances reinforce Russian Foreign Minister Sergei Lavrov's comment Sunday that Assad's removal from power is "impossible to implement," the newspaper said.
Assad's presence on the ship could be a sign of looming negotiations on the conflict in Syria, the report said.
"It is necessary to make everybody, including the opposition, which is still categorically denying any dialogue, to sit down at the negotiating table, Radio Free Europe Radio Liberty quoted Lavrov as saying during a visit to the Ukraine.