TOKYO—Apple Inc. supplier Japan Display Inc. is in advanced talks for a bailout of about ¥75 billion ($703 million) from a government-backed fund that is its largest shareholder, people familiar with the matter said.

The plan under discussion calls for Japan Display to receive new financing from Innovation Network Corp. of Japan, the government-backed fund that holds 36% of the display maker, the people said. Among the options under study is issuance of subordinated bonds by Japan Display to INCJ. Japan Display’s banks also...