The heart of the story--that Warner Brothers wants Affleck, and that they plan on meeting with the director later soon to discuss the possibility, is likely true, according to a report at Deadline Hollywood, but their reporter says that the rumors have been circulating for a while now, and that when he checked in with Affleck's people, the said he wouldn't accept the job.
"This is a story I checked out days ago, and didn’t run when Affleck’s reps stated that it was not going to happen with him," Deadline reports. "Just because the studio wants Affleck doesn’t mean he will do the movie, and several sources tell me he might take a meeting, but that’s it."
The story goes on to imply that Affleck, whose acting career suffered after taking a number of high-profile projects that appeared to be just for the money, would be unlikely to make a Justice League movie unless he had a take on the project that he and the studio thought was better than Joss Whedon's The Avengers. The films he's chosen as a director have done nothing but raise his profile, creating a situation where a studio might trust him with a film this big, and so it follows that if he took the job and couldn't live up to expectations, it could be another decade of reputation rehab, much like the one he just finished.
It's hard to say who's right--Deadline's version of events sounds pretty plausible, but certainly it's hard to imagine many filmmakers walking away from this kind of opportunity.
Maybe Deadline's "several sources" and Variety's "multiple sources" should get together and go bowling.