  We investigate whether a class of models describing F-theory
compactifications admits a specific type of flux vacua with an exponentially
small vacuum expectation value of the superpotential, by generalizing a method
recently developed in Type IIB flux compactifications. First we clarify that a
restricted choice of $G_4$-flux components reduces a general flux
superpotential into a simple form, which promotes the existence of
supersymmetric vacua with one flat direction at the perturbative level. Then we
utilize the techniques of mirror symmetry to determine one-instanton
corrections to the potential and investigate in detail the vacuum solutions of
a particular model.
