The move comes as Prime Minister Narendra Modi's ruling alliance faces a tightening race for an election due by May, with opinion polls suggesting it could fall short of a parliamentary majority.
The government, under pressure from its Hindu base to build a temple to Rama on a site where zealots demolished a 16th century mosque, said land around the disputed site could be given to the trust while the court decided the title suit.
After the 1992 destruction of the Babri mosque that triggered Hindu-Muslim riots across India which killed at least 2,000 people, the Supreme Court ordered a freeze on activity at the disputed site and surrounding areas.
But it has not moved forward in resolving the dispute, prompting a renewed campaign by Hindu hardliners for the construction of the temple at the site they believe to be the birthplace of Rama.
Leaders of Modi's Hindu nationalist Bharatiya Janata Party, hope the handover will help assuage the hardliners, who can begin some form of construction ahead of the election.
"We are trying to do it in the legal way," said Subramaniam Swamy, one party leader, adding that if the Supreme Court allowed the surplus land to be given to the Ram Janambhoomi Trust, it could start construction of the temple.
But one leader of India's tiny Muslim minority, which makes up 14 percent of a population of 1.3 billion that is 80 percent Hindu, said the government's latest proposal on the Ayodhya dispute was aimed at shoring up its base.
The BJP lost power in three states in assembly elections in December, and wants to avoid a similar result during the general election in Uttar Pradesh, a state of 220 million people where Ayodhya is located.