Al Masirah TV has quoted an unnamed Houthi spokesperson as saying that the movement had targeted Saudi Aramco's facilities located in Jizan near the Red Sea.
The spokesperson provided no timeframe for the attacks, with Aramco declining to comment on the report.
The Houthis specifically targeted the Abha and Jizan airports, as well as Khamis Mushait military base and other facilites in Saudi Arabia "with a large number of rockets and drones", according to the spokesperson.
The report comes after Reuters quoted independent UN monitors as saying earlier in January that “despite their claims to the contrary, the Houthi forces did not launch the attacks on Abqaiq and Khurais on 14 September 2019”.
Two Saudi Aramco plants - in Abqaiq and Khurais - were attacked by drones at the time, leading to massive fires and a suspension of the production of 5.7 million barrels of crude oil per day.
The armed Houthi political opposition faction in Yemen earlier claimed responsibility for the attacks.
CNN cited a spokesman for Yemen's Houthi faction, Brig. Gen. Yahya Saree, as saying that the Houthis had attacked Saudi Aramco's largest oil facilities by planning "an accurate intelligence operation and advance monitoring and cooperation of honorable people inside the kingdom".
US State Secretary Mike Pompeo was quick to point the finger at Iran over the attacks, accusations that were vehemently rejected by Tehran.
Houthi forces previously carried out a drone attack on Riyadh's Shaybah oil field and refinery, prompting a counter-attack by the Saudis on targets in northern Yemen.
Yemen Crisis Shows No Sign of Ceasing
The developments came amid the armed conflict in Yemen between government forces led by the exiled President Abdrabuh Mansour Hadi and the Houthis which has been ongoing since 2015.
A Saudi-led coalition has been carrying out airstrikes against the Houthis at Hadi's request since March of that year.
The United Nations has repeatedly called the conflict in Yemen the world’s worst humanitarian crisis, with an estimated 24 million people - more than 80 percent of the country’s population - currently in need of aid.