Gopal Singh Visharad filed the application. His counsel and senior advocate P.S. Narasimha mentioned the matter before a bench headed by Chief Justice Ranjan Gogoi and Justices Deepak Gupta and Aniruddha Bose.
Narasimha said an application had been filed by the plaintiff. He contended that not much progress had been made in the first round of mediation process, which was initiated by the top court.
The court allowed him to file the application and said it will see to his plea for listing.
On May 10, the Supreme Court extended the term of the mediation committee on the Ayodhya dispute till August 15.
The court said the members of the committee were not experiencing any difficulty in the mediation process.
It observed that the Chairman of the committee, former Supreme Court judge F.M.I. Kalifulla, had indicated progress in mediation so far and sought extension of time to complete the task.
The court refused to divulge the details on the mediation efforts.
"We will not tell you the progress made, that is confidential," replied the court to a counsel's query regarding the mediation. The court observed that the mediation process can be also extended, if required.
The court's observation came after a report was submitted on the mediation efforts.
Senior counsel Rajeev Dhavan, appearing for one of the Muslim litigants, told the court that they have no objection to the mediation process taking time.
Iqbal Ansari, the son of Muslim litigant Hashim Ansari in the Ayodhya case, said: "This matter is very old and the issue cannot be resolved in two months."
Iqbal Ansari said that an eventual decision should be such that brings peace to the country.
Swami Satyendra Das, the main priest at the Ram temple in Ayodhya, said the mediation team had submitted its report in the Supreme Court which will decide on the matter.
Das told : "The meditation team has spoken to all the parties and come to a conclusion and given it to the court. Now the court will decide on the matter."
The top court had on March 8 set up a three-member committee headed by Kalifulla to mediate to find a possible settlement of the Ram Janmabhoomi-Babri Masjid Ayodhya title dispute.
The Kalifulla panel has Art of Living founder Sri Sri Ravi Shankar and senior advocate Shriram Panchu as its other members.
The mediation is taking place at Faizabad, close to Ayodhya.
The other judges on the Supreme Court bench were Justice S.A. Bobde, Justice D.Y. Chandrachud, Justice Ashok Bhushan and Justice S. Abdul Nazeer.
( With inputs from IANS )