The Greek-flagged Sounion is being towed northwards under military escort, the source said.
The tanker was hit last month by Iran-backed Huthis off the coast of Hodeida while carrying 150,000 tonnes of crude oil.
"(The tugboat) Aigaion Pelagos has begun to gradually tow the tanker northwards, escorted by military vessels," the defence source told AFP, adding that the ships' radars had been turned off for security reasons.
The European Union's Red Sea naval mission, Aspides, had earlier said that "the salvage operation of the MV SOUNION is essential in order to avert a potential environmental disaster in the region."
"To achieve this, several public and private actors are working together," it added.
The Greek-flagged oil tanker Sounion was hit by missiles off the coast of Hodeida on August 21.
Damage to the vessel had threatened a Red Sea oil spill four times the size of the 1989 Exxon Valdez disaster off Alaska.
The Sounion's crew, made up of 23 Filipinos and two Russians, was rescued the day after the attack by a French frigate serving with the EU mission.
The EU naval force was formed in February to protect merchant vessels in the Red Sea from attacks by the Huthi rebels, who have waged a campaign against international shipping that they say is intended to show solidarity with Hamas in its war with Israel in the Gaza Strip.
Since November, the Huthi attacks have resulted in the sinkings of two ships and the deaths of at least four crew members.
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