Cold dark matter models predict the existence of a large number of substructures within dark matter haloes. If the cold dark matter consists of weakly interacting massive particles, their annihilation within these substructures could lead to diffuse GeV emission that would dominate the annihilation signal of the host halo. In this work we search for GeV emission from three nearby galaxy clusters: Coma, Virgo and Fornax. We first remove known extragalactic and galactic diffuse gamma-ray backgrounds and point sources from the Fermi 2-yr catalogue and find a significant residual diffuse emission in all three clusters. We then investigate whether this emission is due to (i) unresolved point sources, (ii) dark matter annihilation or (iii) cosmic rays (CR). Using 45 months of Fermi-Large Area Telescope (Fermi-LAT) data we detect several new point sources (not present in the Fermi 2-yr point source catalogue) which contaminate the signal previously analysed by Han et al. Including these and accounting for the effects of undetected point sources, we find no significant detection of extended emission from the three clusters studied. Instead, we determine upper limits on emission due to dark matter annihilation and CR. For Fornax and Virgo, the limits on CR emission are consistent with theoretical models, but for Coma the upper limit is a factor of 2 below the theoretical expectation. Allowing for systematic uncertainties associated with the treatment of CR, the upper limits on the cross-section for dark matter annihilation from our clusters are more stringent than those from analyses of dwarf galaxies in the Milky Way. Adopting a boost factor of ∼103 from subhaloes on cluster luminosity as suggested by recent theoretical models, we rule out the thermal cross-section for supersymmetric dark matter particles for masses as large as 100 GeV (depending on the annihilation channel).