Sources said the government had planned to bring the bill in Raj ya Saba May 5, first day of the last week of Budget session.
It is learnt that the issue also came at a meeting of the Business Advisory Committee of the Rajya Sabha today, where Congress opposed the government's move.
While the government has got various states, including Trinamool Congress-ruled West Bengal, on board on the issue, its decision to keep Assam out of it has irked Congress which rules that state.
A senior government functionary had said that since the issue of land swap under LBA with Bangladesh has evoked "emotional reaction" in Assam, the bill will not include any land pertaining to the state.
BJP unit in Assam has opposed the exchange of enclaves between Bangladesh and India involving the territory of Assam.
Sources said that Congress today conveyed to the government that it will oppose the bill unless Assam is included.
"How can you keep Assam out of it? The government of Assam was in favour of the bill. Assam Chief Minister Tarun Gogoi had even accompanied Prime Minister Manmohan Singh when he had visited Bangladesh in September, 2011 to sign an enabling agreement to implement LBA when UPA was in power.
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