The state has accused Exxon of holding back on drilling while it pursues other ventures and has tried to regain control the 106,000-acre field, the Anchorage Daily News reported Thursday.
Exxon is the dominant leaseholder of the oil field, the newspaper said.
The argument went to court Wednesday. Mark Ashburn, representing the state, told Judge Patrick McKay that a suit on the issue was moot, as the state had not yet given Exxon its final decision on a permit to start drilling.
Exxon said it was "ready, willing and able," to initiate its drilling in the field that could produce 10,000 barrels daily of condensate -- a light oil -- by 2014, the newspaper reported.
Ashburn called Exxon's assertion that it will drill soon "a public relations ploy."
The state contends the company is using the field as a natural warehouse and does not intend to begin drilling soon.
A decision on whether or not Exxon can continue with the suit could come within a week, Ashburn said.