Aurinia's shares closed up nearly 27% on the news, giving it a market capitalization of $3.59 billion, according to Refinitiv data.
The report said no final decision has been made and that Bristol Myers could opt out of a potential deal.
Both Bristol Myers and Aurinia declined to comment on the matter.
Aurinia sells a treatment for a kidney disorder called lupus nephritis and is testing out treatments for other autoimmune and kidney diseases, according to its website.
(Reporting by Amruta Khandekar and Ruhi Soni; Editing by Devika Syamnath)