ArcelorMittal, the world's largest steelmaker, has made a $1.5 billion bid for ThyssenKrupp's plant in Alabama, while Brazil's CSN is offering $3.8 billion for both that plant and a majority stake in Brazilian mill Cia. Siderúrgica do Atlántico SA (CSA), the source said.
The Wall Street Journal had earlier reported details of the bids from ArcelorMittal and CSN.
The Wall Street Journal report cited people familiar with the situation as saying US steelmaker Nucor Corp also offered $1.5 billion for the Alabama plant but was less likely to succeed.
The two mills that make up Thyssen's Steel Americas business were meant to carve out new markets for Germany's biggest steelmaker but have suffered from cost overruns, poor project management and weak demand for steel due to the economic downturn.
Thyssen has had to write down the value of the mills, which lost about 1 billion euros in the financial year ended September 2012, by almost half to 3.9 billion euros.
Two people close to the deal said final bids for the mills were due at the end of February.
ThyssenKrupp declined to comment on the report.
ArcelorMittal, which said last week that it had put in a bid to buy assets from Thyssen, also declined to comment.
The German company's shares rose 2 percent to 18.36 euros by 1034 GMT, outperforming the market, as the details of the bids helped to allay concern that the company might fail to find any buyers.
"Given the early stage of the bidding process, we might see that those first offers may be only the start for ThyssenKrupp and that especially for the U.S. asset we see some upward adjustments," UBS analyst Carsten Riek said.
He said that ThyssenKrupp could face another writedown if it accepted CSN's bid.
Thyssen holds its annual shareholder meeting on Friday, where Chairman Gerhard Cromme is expected to come under fire for allowing the disastrous Steel Americas project to continue for as long as it did.
Over the years, Thyssen has invested about 12 billion euros in the mills but then had to write down their value because bids for Steel Americas were lower than expected, causing a huge loss for 2012.
The book value figure of 3.9 billion euros includes the 27 percent of CSA owned by Brazil's Vale, the world's No.2 miner.
Thyssen has said the Steel Americas project will be sold this year, as it shifts investment into higher margin products like elevators, plant components and submarines.