The News of the World reports that the DJ, who currently hosts the early afternoon slot, is considered to be the future of the station by BBC bosses and will take on the role once Moyles has served out another contract.
The news is said to be "devastating" for fellow DJ Scott Mills, who has apparently wanted the breakfast job for years.
"Greg is the future of Radio 1," a source said. "He's in his 20s, cool and a star on the rise. Unfortunately for Scott, he's just too old to get the breakfast gig now because the station is meant to be targeting teenagers and people under 24.
"Greg is the obvious choice for the job, but we have to be sure that he is ready. He is most definitely the heir apparent and it means we will soon have to have a discussion with Chris about how long he will be willing to stay on for.
"We are going to be careful not to force Chris out before Greg is ready. We're well aware Chris is a huge star, but we have to look to the future because someone who's nearly 40 can't really be hosting a show aimed at teenagers."