The Congress is understood to have offered the Left parties the deputy primeministership as well as the home and agriculture portfolios in an attempt to soften them.
The Congress has also reassured the Left that it will not confront them on issues about which they are most sensitive: undermining subsidies, the World Trade Organisation and privatisation.
The Left parties also have reservations about the burgeoning defence relationship between India and the United States, which involves joint military exercises, and with the emergence of Israel as the largest supplier of defence equipment to India.
They would like these reservations to be addressed, something the Congress will have to mull over, given sovereign understandings.
Communist Party of India (Marxist) General Secretary Harkishen Singh Surjeet has reportedly conveyed the Congress' stance to his party, apparently in an attempt to try and influence it into joining the government.
However, the party itself is still in the process of deciding whether to join the government or support it from outside.
In an attempt to get broader backing for his position that the Left would lose a historic opportunity by not joining the government this time, Surjeet tried to rope in the Communist Party of India for a joint meeting on Friday, and proposed this be followed by another meeting so that a common strategy could be evolved.
The Kerala and West Bengal Assembly elections in 2006 are causing the Left parties most anxiety. Apart from the obvious contradiction in supporting the Congress at the Centre and opposing it in the states, the Left has another fear that it does not want to express openly: that should Sonia Gandhi's foreign origins become an issue, the Left will have to answer to the charge of not being nationalistic.
This could cause a polarisation between the Left-Congress combine on the one side and the Bharatiya Janata Party on the other and take away the Left's opposition space.
Till late evening the central committee of the CPM, which is trying to evolve consensus on joining a Congress-led government, could not reach it.
In the afternoon, senior CPM leader Jyoti Basu told reporters that only four members had spoken and 23 others were yet to express their views. The meeting discussed issues ranging from the economic agenda, divestment and joining a Sonia Gandhi-led government.
CPI M sources said a decision on whether to participate in the coalition government was expected to be taken on Monday by the Central Committee.
As many as 70 members were present at the party Central Committee meeting.
Meanwhile, at least one constituent of the Left Democratic Front, the CPI, said, pending a decision on whether it would joint the government or not, it would submit a letter to President Abdul Kalam to support a Congress-led government at the Centre.
The party's national executive meeting which began on Saturday was considering whether they join the government or extend outside support.
The meeting was held in the backdrop of Kerala leaders V S Achutanandan, P Vijayan and MA Baby opposing the CPI M joining the Congress-led government and Jyoti Basu, Surjeet and Yechuri favouring it, sources said.