The ruling BJP welcomed the apex court's suggestion, insisting the parties to the case should keep in mind its "sensitivity", while the Congress was guarded in its response, saying there should be a "consensus-based" solution or the SC adjudicate the matter on merit.
Union Ministers Giriraj Singh and Uma Bharti, considered Hindu hardliners in the Modi government, insisted that the temple be built at the disputed site.
Union Minister P P Chaudhary said the government favours talks to resolve the Ayodhya dispute.
Muslim organisations, who are key to any amicable out- of-court settlement, however, voiced scepticism over efficacy of talks to resolve the long-standing dispute.
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"We welcome this step and I believe they should have talks outside the court," BJP spokesperson Sambit Patra said, adding the party will study in detail the observations made by the apex court, which has said the issue is "sensitive and sentimental", and is related to faith of millions of people.
Muslim organisations, including the Babri Masjid Action Committee (BMAC), which is a party to the case, were sceptical about the Supreme Court's suggestion, saying attempts at out- of-court settlement in the past have been unsuccessful.
"We are ready with Chief Justice (of India) mediating... We trust him. We are also ready we if he nominates a team for hearing the matter. But out of court settlement is not possible. If SC passes an order in this regard, we will look into it," said Zafaryab Jilani, convenor of the committee.
The Secretary General of All India Muslim Personal Law Board (AIMPLB), Maulana Wali Rahmani, said, "We have trust in the Chief Justice and if he is there we are ready for talks."
"Talks were held earlier also but it can be initiated again. But talks should not be long," he added.
Maulana Khalid Rashid of the AIMPLB alleged that the attempts in the past to resolve the issue were foiled by political parties.
BJP veteran L K Advani, one of the protagonists of the movement whose 'rath yatras' for construction of Ram temple galvanised and polarised Hindus to the advantage of the saffron party, said all concerned should reach a consensus to resolve it.
"The Supreme Court's observation is a welcome step and I hope in the light of the apex court's advise, all concerned parties will reach consensus and will find a solution to resolve the Ram Temple issue," Advani, who was the flag bearer of the Ram Janmabhoomi movement, said.
"We were always ready. Mandir and Masjid should be built but Masjid should be built on other side of Sarayu river. The Ram Janambhoomi should be entirely for Ram Mandir," he said, arguing, "we cannot change the birthplace of Ram, but Masjid can be made anywhere."
Bharti, who was closely associated with the Ram Janmabhoomi movement in the 1990s and an accused in the Babri Masjid demolition case along with Advani, insisted that the issue of ownership of the disputed site be settled through talks. She said whosoever owned the land should offer it to Lord Ram as a "gift".
"There can't be a bigger gift than this...We should arrive at a solution which will be remembered for thousands of years," she said.
Terming the Supreme Court's observation as a "solid" one, Adityanath said, "It is a welcome step."
Vishwa Hindu Parishad (VHP) chief Pravin Togadia said the Centre must bring out a legislation to build the Ram temple.
The Ayodhya issue should be resolved at the earliest and a temple built with the involvement of all Indians, RSS functionary Dattatreya Hosabale said.
He said, "It (Ram temple issue) is to be decided by the Dharam Sansad as they are the people who organised the entire Ram Janmabhoomi movement and the parties who went to the court.