TOKYO - U.S. buyout fund Bain Capital has made a counter-offer to buy Japanese software developer Fuji Soft for about 600 billion yen ($4.1 billion), or around 5% higher than private equity firm KKR's bid, the Nikkei newspaper reported on Tuesday.

Bain is believed to have proposed a price of around 9,200 to 9,300 yen for each Fuji Soft share, against KKR's tender offer last month of 8,800 yen a share, the Nikkei said. The stock closed at 8,970 yen on Tuesday.

Fuji Soft's management has been at odds with some of its major shareholders, including Singapore-based fund 3D Investment Partners.

In March, 3D, Fuji Soft's top shareholder with a stake of some 21%, proposed that the company appoint an outside auditor to ensure it fairly review multiple takeover proposals it had received from global private equity firms.

Bain and Fuji Soft both declined to comment on the report.

($1 = 146.3900 yen)

(Reporting by Satoshi Sugiyama, Chang-Ran Kim and Miho Uranaka Editing by Shri Navaratnam)