Abstract We obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space B n {B_{n}} , and we prove that the size of the space dim ( B n ) {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions d = 2 , 3 {d=2,3} , d = 4 , 5 {d=4,5} and d ≥ 6 {d\geq 6} , respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case λ = μ {\lambda=\mu} of bilinear quasimode estimates improves L 4 {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on L p L^{p} -norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when d ≥ 8 {d\geq 8} . And on this basis, we give approximation bounds in H - 1 {H^{-1}} -norm. We also prove approximation bounds for the products of quasimodes in L 2 {L^{2}} -norm using the results of L p {L^{p}} -estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.