It cited a spokesman for the National Oil Corporation (NOC) as saying that with Friday's reopening of Al-Hariga terminal near Tobruk in the east, "production today has passed the 300,000 bpd mark".
NOC officials were not immediately available to confirm this, but the corporation's website said on Friday exports from Al-Hariga had resumed with the departure of the tanker Seachance, loaded with 660,000 barrels of crude.
The specialised Marine Traffic website said the Malta-flagged vessel departed Al-Hariga on May 20, bound for the French port of Fos-sur-Mer.
LANA said on Tuesday the North African nation's production could soon reach 360,000 bpd.
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Oil is Libya's main natural resource, with reserves estimated at 48 billion barrels, the largest in Africa.
It had an output capacity of about 1.5 million bpd before the 2011 revolt that ousted dictator Moamer Kadhafi, accounting for more than 95 percent of exports and 75 percent of the budget.
But production later slumped amid violence as rival militias battled for control of oil terminals.
The new UN-backed Government of National Accord last month secured the backing of the Tripoli-based NOC as the GNA seeks to assert its control over the country.
The GNA plans to focus on the country's key oil sector after the country's economy was hard-hit in the wake of the 2011 uprising.