TOKYO (Reuters) - U.S. buyout fund Bain Capital has made a counter-offer to take over Japan's Fuji Soft for about 600 billion yen ($4.10 billion), or about 5% higher than U.S. private equity firm KKR's bid, the Nikkei business daily reported on Tuesday.

KKR last month launched a tender offer for the Japanese company for 8,800 yen a share. Bain is believed to have proposed a price of around 9,200 to 9,300 yen for each Fuji Soft share, which closed at 8,970 yen on Tuesday.

($1 = 146.3900 yen)