Baring said in a statement it has made a friendly offer of 2,000 yen a share, higher than Bain's offer of 1,670 yen and valuing Nichiigakkan at 146 billion yen ($1.37 billion). It said the price was indicative and that it was open to negotiation.

Nichiigakkan shares closed at 1,698 yen, up 2%, on Monday after rising almost 5%.

Baring's proposal could put pressure on Bain to further sweeten its offer amid a government push for better corporate governance, with boards increasingly expected to seek better returns for minority shareholders.

"BPEA has sent letters to certain members of Nichiigakkan's founding family, offering to evaluate assisting with a management buyout as well as a tender offer for all outstanding ordinary shares and share acquisition rights of Nichiigakkan," Baring said.

Hong Kong-based investment firm Lim Advisors, which holds an undisclosed stake in Nichiigakkan, asked Nichiigakkan's management on Monday to seek an extension on Bain's offer and reiterated fair value for the Japanese company was 2,400 yen a share.

Tokyo-based Nichiigakkan's core businesses of elderly care, childcare and hospital consultancy are growing steadily, according to the company, as demand for the services are rising in aged Japan.

Nichiigakkan said in a statement on Monday it has not been contacted by Baring regarding a counter bid, which was earlier reported by the Nikkei Business magazine.

Baring's announcement comes on the day Bain's bid is set to expire. Bain launched its bid in May and extended it as Nichiigakkan shares traded above Bain's offer price. Earlier this month, Bain raised its offer to 1,670 yen a share from 1,500 yen.

The U.S. buyout fund has secured support from Singapore-based Effissimo Capital Management, which owns 12.46% of Nichiigakkan. It said Effissimo would tender its shares in exchange for an undisclosed stake in a vehicle that would indirectly own Nichiigakkan.

Bain also said it has secured a 44% stake from Nichiigakkan's top management and founding family members.

Bain representatives were not immediately available for comment.

($1 = 106.4900 yen)

By Junko Fujita