Nationalised and under special administration, Ilva's main facility at Taranto is notorious as one of the most polluting industrial facilities in Europe but also an economic lifeline in a region with high unemployment.
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But any new deal is likely to mean cuts among the 14,200-strong workforce.
The AcciaItalia consortium includes specialist steel maker Arvedi, the Italian state's CDP investment bank, and Delfin, the holding group of businessman and Luxottica chairman Leonardo Del Vecchio.
Its new proposal to the Italian government late on Friday was a raised purchase price of 1.85 billion euros ($2.087-billion), a commitment to immediately employ 9,800 workers, and invest 3.1 billion euros, the Italian media reports said.
The original bid had envisaged a headcount of 7,800 by next year, rising to 10,300 by 2023, which union boss Maurizio Landini said was unacceptable.
Italy's Minister for Economic Development Carlo Calenda has said a decision will be taken soon.
A second bidder, global steel giant ArcelorMittal's partnership with Italy's Marcegaglia, wants to cut staff to 8,400 by 2023, according to union officials. However it has promised to invest 2.3 billion euros in Ilva in addition to a reported purchase price of almost two billion euros.
Ilva was nationalised and placed under special administration in 2015 after the Riva family, which owned it, was accused of failing to prevent toxic emissions from its Taranto site, in southern Italy.
The government quickly opened a tender for the steelworks which used to produce a third of the nation's output.