At the court hearing on Wednesday the judges closely questioned both sides but gave no firm indication of how they would rule.
However in previous cases they generally have imposed limits on huge awards of punitive damages imposed on corporate defendants.
ExxonMobil recently posted the highest ever quarterly profits for a US company at more than $11 billion.
A central issue in the case is whether the company can be held liable for the mistakes of Hazelwood, who violated company rules when the Exxon Valdez ran aground in Alaska’s Prince William Sound in March 1989, spilling about 11 million gallons of crude oil.
ExxonMobil afterwards spent about $2.1 billion dollars cleaning up
the polluted coastline and more than $300 million in compensation for
fishermen and locals affected by the catastrophe.
The company also paid out more than $900 million in fines in a bid to halt criminal proceedings begun against it by the US government and the state of Alaska.
However in in 1994, a jury in a civil Alaskan lawsuit ordered the firm to pay five billion dollars in damages to fisherman and locals whose livelihoods were destroyed.
The oil giant has fought that verdict ever since and in 2006 won a victory from a US federal court which cut the punitive damages to $2.5 billion.
Walter Dellinger, a lawyer for Exxon Valdez, said the company could not be held responsible for Hazelwood’s actions as he was not a managerial employee who sent company policy.
“Exxon gained nothing by what went wrong in this case and paid dearly for it,” he said in court.
However Jeffery Fisher, representing the plaintiffs, said the company had failed in its responsibilities.
“Up and down the corporation … for three years, upper management was receiving reports that this man was drinking aboard the vessel,” Fisher told the court.
Samuel Alito, a US supreme court judge, excused himself from the case because he owns ExxonMobil stock, therefore only eight court members heard the case.
Should the judges have a split vote, then an earlier appeals court’s ruling that upheld the award against ExxonMobil would be simply affirmed.