An Israeli naval commander told Aljazeera his ship received an SOS distress call on Wednesday evening.
"As soon as we received the call we sent an emergency vessel to look for survivors," Yusi Mashita, commander of an Israeli marine missile unit, said.
"We believe it sank because of overloading the ship."
Three members of a total of seven crewmen on board the ship have been rescued by a Turkish cargo vessel travelling along the busy sea-lane 64km off the Israeli coast.
A British military helicopter airlifted the three men to another Syrian ship heading to Syrian waters.
The fate of another four members of the sunken vessel is unknown.
The Israeli navy says it dispatched vessels and helicopters to search for the missing crew, but search-and-rescue operations were suspended after several hours due to darkness and rough maritime conditions.
The search will resume at daybreak on Thursday, an Israeli military spokesman said.