I was typing up my response on why I thought he wouldn't do it, but then I did the math and it seemed pretty attainable.
LeBron needs 13,474 more points to catch Kareem. If he plays 10 more seasons, and does the following, he'll be just about there (the following gets him 13,424):
Age 31, Games Played 72, PPG 23
Age 32, Games Played 72, PPG 23
Age 33, Games Played 72, PPG 23
Age 34, Games Played 72, PPG 23
Age 35, Games Played 68, PPG 20
Age 36, Games Played 68, PPG 20
Age 37, Games Played 68, PPG 18
Age 38, Games Played 68, PPG 16
Age 39, Games Played 68, PPG 14
Age 40, Games Played 68, PPG 12
I think it's a pretty safe assumption to say LeBron's never playing 82 games again (actually he's never played 82 games in the first place). Even if 100% healthy, he's taking some games off every season.
We'll see consistent strong scoring over the next 4 seasons, but maybe with some decline as he tries to facilitate more and lets Love and Irving (or whoever) take more of the scoring load.
Age 35-36 will probably be that drop off, when he's still an All-Star, but not automatically All-NBA. And as he gets older he'll need more rest and probably incur a few more at least minor injuries, so we're dropping a few points and games from his averages.
Then if he sticks around, age 37 is when we'll see a more noticeable decline in production each year.
I like to think those assumptions are reasonable, though some may disagree.