Director and film critic John Campea revealed on his YouTube show, "I'm hesitant to say anything, but I talked to a Hollywood producer yesterday who told me who Matt Reeves wants as Batman." Campea goes on to say that he was taken aback by the news and even called a source close to the project that confirmed what he had heard.
According to Campea, Matt Reeves wants Jake Gyllenhaal to step up and play Bruce Wayne. It's important to note that this is an unconfirmed rumor at this time, but even if Gyllenhaal were to be announced, it wouldn't be until after the Justice League dust has settled.
Jake Gyllenhaal was up for the part of Batman in Christopher Nolan's Dark Knight trilogy, but the part ultimately went to Christian Bale (which is good, since Jake's sister Maggie ended up playing Bruce Wayne's love interest in The Dark Knight Rises). It would be a neat turn of events to see Gyllenhaal get his second chance and give his take on the character.