Using the theory of fixed point index, the existence of solutions for the singular second-order boundary value problems x′′(t) + f(t, x(t), x′(t)) = 0, 0 < t < 1, δx(0) = γx′(0), x(1) = αx(η), 0 < α < 1, 0 < η < 1, δ > 0, γ ≥ 0, is presented, where f(t, x, y) may be singular at x = 0.