The alleged new X-Men director is someone who knows the franchise well.
According to Collider, which cites multiple sources surrounding the franchise, the person who may be taking over behind the camera on the next X-Men movie is Simon Kinberg, the writer and producer who has been involved with 20th Century Fox's Marvel properties -- especially X-Men -- for over a decade.
Although Kinberg has yet to direct a film on his own, he's been a constant presence as a writer and producer on films like X-Men: The Last Stand, X-Men: Days of Future Past, X-Men: Apocalypse, Deadpool, Logan and the upcoming New Mutants. He's also an executive producer on the new X-Men-related TV series Legion and credited as a producer on 2015's Fantastic Four reboot as well as the yet-to-be-made Gambit.
In other words, Kinberg may not quite be the equivalent of Marvel's Kevin Feige or DC's Geoff Johns, but he's been a creative participant in Fox's Marvel mini-universe for quite some time. He knows the characters and the actors and has been on set for many of these productions. Whether that can translate into actual directorial skills is another matter, but he does come equipped with a strong working knowledge of the franchise itself.
Kinberg would succeed Bryan Singer, who kickstarted the X-Men series back in the early 2000s and returned to helm X-Men: Days of Future Past and X-Men: Apocalypse, although his inspiration seemed to wane with that last one.
Those same sources say that Kinberg already has a script completed for the film, which may or may not be titled X-Men: Supernova, and that the film will in fact adapt the classic "Dark Phoenix" storyline handled so disastrously in X-Men: The Last Stand (maybe Kinberg wants to make amends for that by doing it all himself this time).
Sophie Turner, who plays Jean Grey -- around whom the "Dark Phoenix" plot is focused -- recently said she's "about to start shooting" a new X-Men movie, so it's probable that she means this one and not New Mutants, which is also starting production under director Josh Boone. What we still don't know is whether more of the current X-cast, last seen in 2016's X-Men: Apocalypse, will return.
James McAvoy (Professor X) was supposedly ready to come back, and it's likely that newer cast members like Tye Sheridan (Cyclops), Evan Peters (Quicksilver), Kodi Smit-McPhee (Nightcrawler) and Alexandra Shipp (Storm) will return too. On the other hand, Michael Fassbender (Magneto), Jennifer Lawrence (Mystique) and Nicholas Hoult (Beast) may all be finished with their contracts.
None of this is official for now, but if a direct sequel to X-Men: Apocalypse is really in the works and really going to take another (and hopefully better) crack at the "Dark Phoenix" saga, how do you feel about Simon Kinberg directing it?