The company is expected to finalise the deal as early as the end of the month, Dow Jones Newswires reported, citing unnamed people close to the discussions.
The cash boost, worth around $638 million, would be equivalent to 12 percent of Olympus' current market capitalisation, the report said.
It was one of a number of offers made for the endoscope and camera manufacturer in the wake of the revelation senior figures at the firm had covered up losses dating back to the 1990s.
Olympus is likely to face pressure to announce a capital boost ahead of its November issuance of quarterly earnings, Dow Jones said, adding a decision on Sony's offer was expected to come at the end of September or early October.
Last month Olympus said it was selling its mobile phone unit to a domestic investment fund for $674 million and expected to record this as a one-time gain in the second-quarter financial report.
The company reported a 4.46 billion yen loss in the April to June quarter.
For the fiscal year to March, Olympus has said it expected to book a net profit of 7.0 billion yen on sales of 920 billion yen.
The firm's reputation, and Japan's corporate governance image, were badly damaged after its British former chief executive blew the whistle over $1.7 billion worth of losses that the company had moved off its balance sheet.
The firm initially denied allegations it had used past acquisitions and outsized consultant fees to hide the huge losses, but eventually admitted wrongdoing.
In March, the company and three former senior executives -- including ex-president Tsuyoshi Kikukawa -- were charged over their role in the scandal.
Olympus has since announced a major overhaul that includes cutting about seven percent of its workforce.
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