"The ship was heading southwest along the Gulf of Aden at a speed of 8.2kts when the forward station was struck by a missile. A fire started but was neutralised," Ambrey said in a statement.
A second missile missed and "small boats in the vicinity opened fire on the ship" causing it to change direction to port.
"No injuries were reported," the security firm added.
The United Kingdom Maritime Trade Operations (UKMTO), run by Britain's Royal Navy, said in a separate statment that it was informed of an incident southeast of Aden on Saturday night, and that authorities were investigating.
"Vessels are advised to transit with caution," it said.
The attack comes amid a campaign of drone and missile strikes against Israeli-linked shipping by Yemen's Iran-backed Huthi rebels.
The rebels' attacks, which they say are in support of Palestinians, have prompted some shipping companies to detour around southern Africa to avoid the Red Sea, a route that normally carries about 12 percent of global trade.
Since January, the United States and Britain have launched retaliatory strikes on Huthi targets in Yemen in response to the attacks on shipping.
The strikes have done little to deter the Huthis, who have vowed to target US and British vessels as well as all ships heading to Israeli ports.
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