Summary: We obtain asymptotic expansions for the Gauss hypergeometric function \[ F(a +\varepsilon_1\lambda,\,b +\varepsilon_2\lambda;\,c+\varepsilon_3\lambda;\,z) \] as \(|\lambda|\rightarrow\infty\) when the \(\varepsilon_j\) are finite by an application of the method of steepest descents, thereby extending previous results corresponding to \(\varepsilon_j= 0, \pm 1\). By means of connection formulas satisfied by \(F\), it is possible to arrange the above hypergeometric function into three basic groups. In Part I, we consider the cases (i) \(\varepsilon_1 > 0\), \(\varepsilon_2= 0\), \(\varepsilon_3= 1\) and (ii) \(\varepsilon_1 > 0\), \(\varepsilon_2= -1\), \(\varepsilon_3= 0\); the third case \(\varepsilon_1\), \(\varepsilon_2> 0\), \(\varepsilon_3= 1\) is deferred to Part II. The resulting expansions are of Poincaré type and hold in restricted domains of the complex \(z\)-plane. Numerical results illustrating the accuracy of the different expansions are given.