We study the effects of new physics in double Higgs production at future $e^+ e^-$ colliders. In the Standard Model the chiral limit ($m_e=0$) plays an important role for this process, being responsible for the smallness of the tree-level diagrams with respect to the 1-loop contributions. In our work, we consider the possibility of an enhancement due to the contribution of Standard Model dimension-six effective operators. We show that there are only two relevant operators for this process that are not yet (strongly) constrained by other data. We perform a sensitivity study on the operator coefficients for several benchmark values of energy and integrated luminosity related to the proposed linear colliders such as CLIC, ILC and FCC-ee and we derive expected 95\% CL limits for each benchmark scenario.