The move comes just two days after Griffin left rival Meredith Corp.'s National Media Group where he had been CEO. Time Warner declined to comment on the reports.
Moore spent 32 years at Time Inc., the last eight as chairwoman-CEO. Her current contract expires in December 2012, according to The New York Times. In 2007, Moore shuttered onetime high-flyer Business 2.0 and late last year closed Fortune Small Business.
At Time Inc., second-quarter operating profit increased 50%, to $153 million; revenue for the quarter reached $919 million.