By Kimberley Kao

LY Corp. and Bain Capital have raised their takeover bid for Kakaku.com in an offer valuing the Japanese internet company at over $4 billion, widening the gap with a rival offer from Sweden's EQT.

In a legally binding proposal dated Wednesday, LY and Bain Capital raised their bid for all shares of Kakaku.com to 3,384 yen a share, up from 3,232 yen proposed in May.

That values the company at about 669.36 billion yen, equivalent to $4.12 billion, given about 197.8 million shares outstanding as of the end of March.

LY and Bain Capital said their offer would increase to 3,500 yen a share if a nontender agreement could be executed with KDDI, one of Kakaku.com's largest shareholders. They added that they won't commence the offer unless Kakaku.com expresses an opinion in support of it.

Kakaku.com said Thursday that it will "carefully evaluate which transaction would better contribute to the enhancement of the company's corporate value and the common interests of its shareholders."

Shares of Kakaku.com rose 3.8% in Thursday afternoon trade.

In May, Swedish investment company EQT offered to acquire Kakaku.com at Y3,000 a share.

Kakaku.com operates a group of online platforms, including a price-comparison site, a restaurant-review site and a job-search site.

Write to Kimberley Kao at kimberley.kao@wsj.com