Leonard Green & Partners and CVC Capital Partners disclosed the bid in a Securities and Exchange Commission filing.
In February, BJ's had said it was exploring a sale. In March, Leonard Green had said it was considering a proposal. It already owns a 9.3 percent stake in BJ's.
Leonard Green has been expanding its retail properties. It made a bid for 99 Cents Only Stores in March, bought Jo-Anne Stores and was part of a group that acquired J. Crew this year.
Westborough, Mass.-based BJ's could not be reached for comment. Its shares rose $2.13, or 4.5 percent, to $49.99 in pre-market trading.
