"You please mention this after three months. We will entertain this then," the bench which also comprised justices S A Nazeer and Sanjiv Khanna said.
The BJP leader submitted that he has already won the first round of the litigation in which the Centre accepted the existence of Ram Sethu.
He further said the union minister then had called a meeting in 2017 to consider his demand to declare the Sethu as a national heritage monument but subsequently nothing happened.
The top court referred to pendency of several cases before it and asked Swamy to mention his plea after three months for an urgent hearing.
The BJP leader had earlier raised the issue of declaring Ram Sethu
as a national monument in his PIL against the controversial Sethusamudram Ship Channel project, initiated by the UPA-I government.
The matter had reached the apex court, which in 2007 had stayed the work for the project on the Ram Sethu.
The Centre had later said that it considered the "socio-economic disadvantages" of the project and was willing to explore another route to the shipping channel project without damaging the Ram Sethu.
"That the Government of India intends to explore an alternative to the earlier alignment of Sethusamudram Ship Channel project without affecting/damaging the Adam's Bridge/Ram Sethu in the interest of the nation," the affidavit filed by the ministry had said.
The court had then asked the government to file a fresh affidavit.
The Sethusamudram shipping channel project has been facing protests from some political parties, environmentalists and certain Hindu religious groups.
Under the project, a 83 km-long deep-water channel was to be created, linking Mannar with Palk Strait, by extensive dredging and removal of limestone shoals.
The apex court had on November 13 last year granted six weeks to the Centre to clarify its stand on the Ram Sethu. It had also granted liberty to Swamy to approach the court if the response of the Centre was not filed.