[Antonella]

First: on a batted ball to the out field first reaction of base umpire is basically expected to move toward the infield.
In Two Ump.s System this movement will be toward the circle.
In Three or Four Ump.s System this movement (not always, but most of the times) will be along the foul line, in foul territory.
PRIORITY of base umpire is batter-runner.
Plate Umpire's priority, on the other hand, is fair/foul call on the batted ball.

So my answer is: fair/foul is for PU.
BU could help (if requested), but at the moment that batted ball is 'landing'... he/she is probably checking something else...

JMO

Ciao