The deal would cover the company's potential penalties under the Clean Water Act and payments under the Natural Resources Damage Assessment, the newspaper said, citing sources familiar with the discussions.
It was unclear if the deal has been formally offered to BP. The U.S. Justice Department declined to comment.
A settlement could avert a bruising courtroom battle over the worst ever U.S. offshore oil spill slated to start on Monday in New Orleans, although the trial may begin as the terms of the deal are hammered out.
A settlement would also put a solid number on BP's costs under the Clean Water Act, which range from $4.5 billion to $17.5 billion, as well as potential natural resources damage assessments to the states under the Oil Pollution Act.
"BP doesn't talk about possible offers or negotiations, but I can tell you we are ready for trial and looking forward to the opportunity to present our case starting Monday," BP spokesman Geoff Morrell said when contacted by Reuters.
BP has spent or committed $37 billion on cleanup, restoration, payouts, settlements and fines. That includes an estimated $8.5 billion deal with most plaintiffs and a record $4.5 billion in penalties, and a guilty plea to 14 criminal counts to resolve criminal charges from the Justice Department and civil claims from the U.S. Securities and Exchange Commission.
BP has said it would settle on "reasonable terms," but was prepared to go to trial if the demands were "excessive and not based on reality."
(Reporting by Karl Plume in Chicago; Editing by Vicki Allen and Gunna Dickson)