TOKYO (Reuters) - KKR extended on Thursday its tender offer period for Fuji Soft to Jan. 9 as it battles rival U.S. private equity firm Bain Capital for control of the Japanese software maker.

Its offer price was unchanged, it said in a statement.

KKR now owns about 34% of Fuji Soft after the first stage of its two-part tender offer, mainly through shares tendered by activist shareholders 3D Investment Partners and Farallon Capital Management at 8,800 yen apiece.

Last month, KKR raised its offer price for Fuji Soft to 9,451 yen - 1 yen above Bain's earlier bid - only to have Bain again increase its offer to 9,600 yen last week. Its second-stage tender offer period was set to expire on Thursday.

Fuji Soft's shares closed at 9,810 yen on Thursday, up 0.4% on the day.

(Reporting by Chang-Ran Kim; Editing by Muralikumar Anantharaman)