A host of interested parties have already lined up to stake a claim to the treasure, believed to be in gold and silver.
One of the king's descendants, Navchandi Veer Pratap Singh, said "if gold is really found there, we should get our share."
Uttar Pradesh state authorities, as well as local officials, also said they had a right to the wealth.
"The treasure trove should be used for the development of the state," local lawmaker Kuldeep Senger said. Uttar Pradesh, with a staggering population of 200 million, is one of the poorest and least developed states in India.
Residents of the impoverished Daundia Khera village, who have no access to electricity, said they have long known about the treasure from stories told by their elders.
"Everyone in the village knows about it," said 60-year-old Vidyawati Sharma, who learned the stories from her father-in-law.
Locals have found silver and gold coins in the area in Unnao district, about 80 kilometers (50 miles) southwest of the state's capital of Lucknow, according to the swami's disciple, Om Ji. "No one knew exactly where" the treasure was until the late king visited the swami in his sleep, he said.
However, officials from the Archaeological Survey of India denied that the agency had begun the excavations at the bidding of the Hindu holy man.
"Archaeology doesn't work according to the dreams of a holy man, or anybody else. Archaeology is a science. We are carrying out this excavation on the basis of our findings" at the site, said Syed Jamal Hasan, an agency official.
Authorities have set up barricades against thousands of people who have since thronged to the village in hopes of seeing the treasure, or possibly taking a small piece home. People were offering prayers at the temple within the fort's ruins.
Locals also said they hoped Swami Sarkar's vision turned out to be real, as he "is revered as God in this area because he has done a lot for this place," schoolteacher Chandrika Rani said.
The Supreme Court said Friday that it would consider a petition for the court to monitor the treasure hunt, amid fears that some of the riches could be stolen.
Indian officials are also unearthing and cataloging another treasure trove found two years ago in a 16th century Hindu temple, and have barred the media and public from the excavation site in the southern state of Kerala. The discovery of that treasure, including bagfuls of coins, jeweled crowns and golden statues of gods and goddesses, made the Sree Padmanabhaswamy Temple the richest known religious institution in India. The former royal family that has remained the temple's trustees since India's 1947 independence has said the treasure belongs to the Hindu deity Vishnu, who is also known in the region as Padmanabhaswamy.