Quoting a Kuwait Investment Authority official it did not identify by name, the paper said BP's outgoing chief executive, Tony Hayward, made the offer in a telephone call to the KIA which owns 1.75 percent of BP.
Hayward said the offer was part of an effort to fend off takeover bids by other international oil companies in the wake of the devastating oil leak in the Gulf of Mexico.
The report said Kuwait, the world's fourth largest oil exporter, had no intention of exiting its investment in the oil major.
There was no immediate comment from the sovereign fund to the report. BP officials were also not immediately available for comment.
Last month, another Kuwaiti daily reported that Kuwait may buy some of BP's Middle East and Asian assets including oil fields in Egypt and Yemen due to BP's need for liquidity.
(Reporting by Diana Elias; Editing by Jason Neely)