Nabil Ashur said in an interview with the London-based al-Sharq al-Awsat newspaper that his country was optimistic the supertanker and its crew would be released.
The report was confirmed by Saudi Foreign Ministry spokesman Khalid bin Saud bin Khalid.
"We have signals giving us hope for a positive outcome soon," he said.
He reiterated that the authorities were not in negotiations with the pirates and did not intend to pay a ransom.
The Liberian-flagged oil tanker MV Sirius Star, owned by Saudi Aramco, was seized 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 demand was later dropped to $12 million.
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 in November that pirates had earned up to $30 million in ransoms this year. The Security Council has also adopted a British-proposed resolution on tougher sanctions against Somalia over the country's failure to prevent a surge in sea piracy.
NATO and the EU are to launch on December 8 a joint naval operation, dubbed Atalanta, against piracy near the coast of Somalia. The operation could involve up to 10 warships.