Asharq Al-Awsat newspaper cited a source close to the hostage negotiations as saying that the ransom demand could be cut further to around $6-7 million and expected the incident to be resolved in the next 3-4 days.
The Liberian-flagged oil tanker MV Sirius Star, owned by Saudi Aramco, was seized on November 15 about 830 km (516 miles) southeast of Mombasa, Kenya.
The supertanker is believed to be carrying some 2 million barrels of crude, equivalent to a quarter of Saudi Arabia's daily output. It is the largest ship ever captured at sea.
The pirates had originally demanded $25 million for the release of the Sirius Star and its crew, insisting that the ransom be paid within the next 10 days.
The crew of 25, includes nationals from Croatia, Britain, Philippines, Poland and Saudi Arabia.
According to Somali negotiators involved in the hostage talks, all the crewmembers are in a satisfactory condition.
Somali pirates have attacked around 90 ships so far this year, resulting in the seizure of around 39 vessels, including 200 crew members. The East African nation has been without a functioning government since 1991.
UN Secretary-General Ban Ki-moon said in his quarterly report to the UN Security Council on Wednesday that pirates had earned up to $30 million in ransoms this year.
The UN Security Council adopted last Thursday a British-proposed resolution on tougher sanctions against Somalia over the country's failure to prevent the surge in sea piracy.
NATO and the EU have recently announced plans to increase their naval presence in the Gulf of Aden and will launch on December 8 a joint naval operation, dubbed Atalanta, against piracy near the Somali coast. The operation could involve up to 10 warships.
In October, Russia sent the Neustrashimy (Fearless) missile frigate from the Northern Fleet to Somalia's coast to protect Russian vessels in the country's waters. The Russian warship has prevented the seizure of at least two foreign vessels in the area.