They are now meeting on May 23 to consider the impassioned plea of Foreign Minister Pranab Mukherjee that he made at today's meeting of the UPA-Left committee on the deal.
The next UPA-Left meeting is scheduled for May 28.
Mukherjee told the Left parties that the "possibility of signing the deal with the US is virtually over because of paucity of time". However, India could not go ahead with clinching similar agreements with Russia or France unless it signs the safeguards agreement with the IAEA, he said.
"I am going to Russia next month. If I have a categorical assurance that I can sign the safeguards agreement, I will take the nuclear negotiations with Russia forward", Mukherjee told the Left parties.
The safeguards agreement means the IAEA will be able to monitor those Indian nuclear installations that the country will classify as civilian. India's military programme will continue unobstructed. This is a major concession as India is not a signatory to the nuclear Non-Proliferation Treaty.
An indication of softening of the Left parties was their decision to meet on May 23 to consider the government's suggestion. CPI(M) leader Prakash Karat reiterated the Left's objection that the IAEA's clearance would mean the matter would go into the "auto-pilot mode" to the next level