Lucchini, formerly owned by Russia's Severstal, was declared insolvent in 2012 and later placed under "special administration" - a procedure designed to save large companies and avoid heavy job losses. It fell victim to the 2008 recession that has cut Europe's steel demand by about a quarter.
"There hasn't been a formal expression of interest yet but in the last few days ArcelorMittal has been carrying out a sort of due diligence of the Piombino," said Gianni Venturi, national co-ordinator for Italian union FIOM CGIL.
Both ArcelorMittal and Lucchini declined to comment when contacted by Reuters.
Also in the running for Lucchini are India's Jindal Steel and Power and JSW Steel, though they are not interested in the company's blast furnace in Piombino, which employs around 4,000 people directly and indirectly.
"Should ArcelorMittal go ahead with a formal offer, this is likely to include either all assets or only the blast furnace since there is already consolidated interest for the processing lines from the Indian Group Jindal South West (JSW)," said Venturi.
Italy's unions and politicians favour an offer that includes a commitment to keep the blast furnace running. The furnace, idled in April, is part of a complex that last year produced 1 million tonnes of steel or 4 percent of Italy's output.
Last month, ArcelorMittal said the Italian government had asked it to consider investing in or buying steelmaker Ilva - the biggest producer in Europe which is also under special administration.
According to Venturi, ArcelorMittal is carrying out due diligence on Lucchini in parallel with its assessment of Ilva.
Italian media reports have said heavy liabilities and a need to protect jobs mean Lucchini could be sold for just one euro. Italian steel industry body Federacciai estimates Ilva is losing money at a rate of between 60 million euros ($81.8 million) and 80 million a month. ($1 = 0.7331 Euros)