Embattled air bag supplier Takata Corp. is moving forward with an auction of the company as private-equity firms and car parts-makers line up offers.

Suitors include Carlyle Group LP, which is working with Chinese-owned air bag manufacturer Key Safety Systems Inc.; Daicel Corp., a Japanese manufacturer of air bag inflators that is jointly bidding with Bain Capital; and KKR & Co., according to people familiar with the matter.

Bidders for Tokyo-based Takata have been asked to submit their proposals by early this week, and Takata aims to shortlist two to three candidates by October, said the people, who asked not to be identified because the information is confidential.