  We obtain Nekrasov-type expressions for the Seiberg-Witten prepotential for
the six-dimensional (1,0) supersymmetric E-string theory compactified on T^2
with nontrivial Wilson lines. We consider compactification with four general
Wilson line parameters, which partially break the E_8 global symmetry. In
particular, we investigate in detail the cases where the Lie algebra of the
unbroken global symmetry is E_n + A_{8-n} with n=8,7,6,5 or D_8. All our
Nekrasov-type expressions can be viewed as special cases of the elliptic
analogue of the Nekrasov partition function for the SU(N) gauge theory with
N_f=2N flavors. We also present a new expression for the Seiberg-Witten curve
for the E-string theory with four Wilson line parameters, clarifying the
connection between the E-string theory and the SU(2) Seiberg-Witten theory with
N_f=4 flavors.
