This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu (SIAM J Numer Anal 33:2417–2430, 1996). We provide a geometric Gauss–Newton method for solving the least squares inverse eigenvalue problem. The global and local convergence analysis of the proposed method is established under some assumptions. Also, a preconditioned conjugate gradient method with an efficient preconditioner is proposed for solving the geometric Gauss–Newton equation. Finally, some numerical tests, including an application in the inverse Sturm–Liouville problem, are reported to illustrate the efficiency of the proposed method.