The deal could lead to foreign activist shareholders being bought out after years of tension.
A consortium led by private equity firm Japan Industrial Partners has been given preferred bidder status in the second round of bidding, a source familiar with the matter told Reuters.
The $19 billion (2.8 trillion yen) bid figure, cited by the Kyodo news agency, would mark a 26% premium of Wednesday's closing price.
Japan Industrial Partners has contacted multiple Japanese firms, sources said.
Orix Corp plans to join the consortium, with Chubu Electric Power Co also considering it, said another source.
Japan Industrial Partners declined to comment.
The fund has been involved in corporate carve-outs and spin-offs from Japanese conglomerates.
It bought Olympus Corp's camera business last year and Sony Group's laptop computer business in 2014.
However, Toshiba and activist shareholders have been at odds over the direction of the company, with several large foreign funds pushing the conglomerate to consider private equity bids.
Toshiba shares were up 7% in afternoon trade, on track for their biggest one-day gain in more than a year.
They have risen around 17% this year.
