According to party sources, the core committee, including Prime Minister Manmohan Singh and party chief Sonia Gandhi, will meet again next week to further review the issue.
The core group discussed whether to bring an ordinance or to debate the bill in parliament.
The bill, Sonia Gandhi's pet legislation, is stuck in parliament as the Bharatiya Janata Party (BJP) did not allow it to be debated in the Lok Sabha during the budget session, which ended May 8. The BJP wanted the Prime Minister Manmohan Singh to resign over the faulty allocations of coal mines.
The core group also discussed the vexed issue of creating a separate Telangana out of Andhra Pradesh, said the sources.
The Congress has lost three leaders over the issue, who have joined the Telangana Rashtra Samithi.
Chances of government formation with the help of Jharkhand Mukti Morcha (JMM) in Jharkhand, which is under president's rule, also came up for discussion at the core group but no decision was taken, said the sources.
Earlier in the day Jharkhand state unit chief Sukhdeo Bhagat and leader of Congress legislative party met party general secretary in charge Shakeel Ahmed.
Congress MLAs from the tribal state have earlier suggested to the central leadership to form a government with JMM support but there are different opinions on the issue, said party sources.