According to people familiar with the matter, the talks are "promising," but nothing has been set in stone.
The publication says that executives from Apex are visiting representatives of Lexmark in the United States this month to try and reach an agreement on terms.
However, Apex has already signed up with a US investment bank and some Chinese financial institutions to fund the deal.
Lexmark, alongside other printer specialists, has suffered in recent times as consumers shift more towards mobile devices and rely less on traditional PC setups. In October, the US firm announced plans to explore strategic alternatives after a tumultuous few years, saying that in order to enhance shareholder value, business transformation would be key.
The company decided to focus on the enterprise market, acquiring software firm Kofax last year in a deal worth $1 billion.
However, the business shake-up has arguably not gone as well as hoped. In Q4 2015, Lexmark reported revenue of $969 million in 2015 compared to $1.02 billion in 2014, with earnings per share of -$0.17, in comparison to -$0.37 year-on-year.
Lexmark then submitted itself to an auction, but according to Reuters, this move did not yield a good result.
Should a deal be reached, Lexmark has the option to sell itself in its entirety and shareholders may not completely lose out on their investment. In addition, Apex could be granted a fresh revenue stream in the US market, as well as access to Lexmark's current enterprise clients.
This may be the only way to keep Lexmark running in some form, or alternatively, the US printing firm is also exploring options which include selling off software and hardware assets -- which could result in a streamlined, albeit smaller business.