HESS J1858+020 is a TeV gamma-ray source that was reported to have no clearly catalogued counterpart at any wavelength. However, it has been recently proposed that this source is indirectly associated with the radio source, re-identified as a supernova remnant (SNR), G35.6−0.4. The latter has been found to be middle-aged (∼30 kyr) and to have nearby molecular clouds (MCs). HESS J1858+020 was proposed to be the result of the interaction of protons accelerated in the SNR shell with target ions residing in the clouds. The Fermi Large Area Telescope (LAT) First Source Catalog does not list any source coincident with the position of HESS J1858+020, but some lie close. Here, we analyse more than 2 years of data obtained with the Fermi-LAT for the region of interest, and consider whether it is indeed possible that the closest LAT source, 1FGL J1857.1+0212c, is related to HESS J1858+020. We conclude it is not, and we impose upper limits on the GeV emission originating from HESS J1858+020. Using a simplified 3D model for the cosmic ray propagation out from the shell of the SNR, we consider whether the interaction between SNR G35.6−0.4 and the MCs nearby could give rise to the TeV emission of HESS J1858+020 without producing a GeV counterpart. If so, the pair of SNR/TeV source with no GeV detection would be reminiscent of other similarly aged SNRs, such as some of the TeV hotspots near W28, for which cosmic ray diffusion may be used to explain their multifrequency phenomenology. However, for HESS J1858+020, we found that although the phase space in principle allows such a GeV–TeV non-correlation to appear, usual and/or observationally constrained values of the parameters (e.g., diffusion coefficients and cloud–SNR likely distances) would disfavour it.