Stephanopoulos, 48, currently hosts the Sunday ABC News program, This Week, and is the network's chief Washington correspondent. According to the report, Stephanopoulos, a former political adviser to President Bill Clinton who has been with the network since 1996, hopes to continue This Week even if he gets the new gig.
Sawyer, 63, will move to the ABC's World News Tonight in January to replace Charlie Gibson as anchor. Gibson announced in September that he would retire at the end of this year. Sawyer has hosted Good Morning America for 10 years.
A rep for ABC News would not confirm the report.