US buyout fund Bain Capital said on Tuesday it made a counter-offer to buy Japanese software developer Fuji Soft (FJT.MU), estimated at about 600 billion yen ($4.1 billion) that would exceed a rival offer from equity fund KKR.

Bain said in a statement it made an offer at about 5% over the 8,800 yen per share tender offer made last month by KKR. 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.

Fuji Soft said in a separate release that it received a non-binding offer from Bain in July, and that its board would consider any legally binding proposal.

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.

($1 = 146.3900 yen)

(Reporting by Satoshi Sugiyama, Chang-Ran Kim and Miho UranakaEditing by Shri Navaratnam and Louise Heavens)