Strong accretion shocks are expected to illuminate the warm–hot intergalactic medium encompassed by the filaments of the cosmic web, through synchrotron radio emission. Given their high sensitivity, large low-frequency radio facilities may already be able to detect signatures of this extended radio emission from the region between two close and massive galaxy clusters. In this work we exploit the non-detection of such diffuse emission by deep observations of two pairs of relatively close (≃10 Mpc) and massive ( M 500  ≥ 10 14   M ⊙ ) galaxy clusters using the LOw-Frequency ARray. By combining the results from the two putative inter-cluster filaments, we derive new independent constraints on the median strength of intergalactic magnetic fields: B 10 Mpc  < 2.5 × 10 2 nG (95% confidence level). Based on cosmological simulations and assuming a primordial origin of the B -fields, these estimates can be used to limit the amplitude of primordial seed magnetic fields: B 0  ≤ 10 nG. We recommend the observation of similar cluster pairs as a powerful tool to set tight constraints on the amplitude of extragalactic magnetic fields.