Semilocal convergence of Newton-like methods \(x_{k+1} = x_k - A(x_k)^\#F(x_k)\), \(k \geq 0\) is discussed for solving the nonlinear equation \(\Gamma F(x)=0\). Here \(F\) is a twice \(F\)-differentiable nonlinear operator between Banach spaces and \(\Gamma\) is a bounded linear operator, furthermore, \(A(x)\) is a bounded linear operator approximating \(F'(x)\) and \(A(x)^\#\) a bounded outer inverse of \(A(x)\), i.e., \(A(x)^\# A(x) A(x)^\# = A(x)^\#\). It is an extension of the work by \textit{M. Z. Nashed} and \textit{X. Chen} [Numer. Math. 66, No. 2, 235-257 (1993; Zbl 0797.65047)]. Convergence theorems of (semilocal) Kantorovich-type and Mysovskij-type are proved under hypotheses on the second \(F\)-derivatives of \(F(x)\).