The in-camera sessions are being held in Faizabad, a town near Ayodhya.
Nirmohi Akhara, one of the key petitioners in the Ayodhya case, has appealed to the Supreme Court, seeking modification in the order that referred the matter to a group of mediators. The mediation process, it told the top court, "left a lot to be desired".
The group has objected to more than 25 parties being invited to the mediation sessions and said only the claimants to the disputed land --- the Sunni Waqf Board and the Akhara -- should be part of mediation. The Waqf Board and the Nirmohi Akhara were the only stakeholders, who had no objections to mediation. The rest, including the Uttar Pradesh government, were against it.
The Nirmohi Akhara has also asked the court to shift the venue of mediation from Faizabad, a town near Ayodhya, to a neutral venue and appoint two more judges as part of the panel of mediators.
The decades-old Ayodhya temple-mosque dispute was referred to a three-member panel of mediators earlier this month. A five-judge constitution bench headed by Chief Justice of India Ranjan Gogoi tasked them with talking to all stake-holders and arriving at a solution within eight weeks.
The process, the court said, should be confidential and barred the media from reporting it.
The court's chosen mediators were retired Supreme Court judge Ibrahim Kalifulla, spiritual guru Sri Sri Ravi Shankar and a senior advocate, Sriram Panchu.
The Hindu groups involved in the case had argued that a solution through mediation would not be accepted by a majority of the community. The court had pointed out that it was not appropriate to pre-judge the matter. The case was about "mind, heart and healing" and not land, the court had said.
The dispute involves 2.77 acres of land in Ayodhya, where a 16th Century mosque -- said to have been built by Mughal emperor Babur -- stood.
Hindu groups believe the mosque was built on the ruins of a temple that marked the birthplace of Lord Ram.
The mosque was razed by Hindu activists in December 1992. In the days that followed, 2,000 people died in riots across the country.