The case is known as a "shareholder derivative" action, because shareholders filed the case on behalf of the company against the firm's own executives and board members, the Anchorage (Alaska) Daily News reported Monday.
The case revolves around the explosion on the Gulf of Mexico Deepwater Horizon oil drilling platform in April 2010.
The explosion killed 11 workers and set off a world-record, underwater oil leak that took months to stop.
A similar case was dismissed in Texas, which may cause the one in Alaska to be tossed, as well. The Alaska litigation was put on hold in August to await the results of the Texas case.
But Benny Goodman III, a lawyer for the shareholders, is arguing the case should continue because BP's track record in Alaska shows a history of environmental mishaps that have harmed the state and its residents.
Goodman, by example, said one of his clients, Jeffrey Pickett, lives in Anchorage and has been involved in efforts to encourage the company to improve its corporate practices, the newspaper said.
Richard Pepperman II, an attorney for BP's board, is arguing that a case of this nature should be tried in Britain, where BP has its headquarters.
Judge Sharon Gleason, who put the case on hold in August, has moved on to a federal bench. She has been replaced in the Superior Court case by Brian Clark, who heard arguments on the merits of the case last week.
Notable deaths of 2014 [PHOTOS]