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We study the globular cluster (GC) system of the Virgo giant elliptical galaxy NGC 4365, using new wide-field K-band imaging from the ESO 3.5m New Technology Telescope, archive V and I imaging from FORS1 on the ESO VLT and HST/WFPC2+ACS data. As in most other large ellipticals, the GC colour distribution has (at least) two peaks, but the colours of the red GCs appear more strongly weighted towards intermediate colours compared to most other large ellipticals and the integrated galaxy light. The intermediate-color/red peak may itself be composed of two sub-populations, with clusters of intermediate colours more concentrated towards the centre of the galaxy than both the blue and red GCs. Nearly all GCs in our sample fall along a well-defined narrow sequence in the (V-K, V-I) two-colour diagram, with an offset towards red V-K and/or blue V-I colours compared to simple stellar population models for old ages. This has in the past been interpreted as evidence for an intermediate-age population of GCs. The offset is however seen for nearly all metal-rich clusters within the 5'x5' SOFI field, not just those of intermediate colours. We combine our VIK data with previously published spectroscopy resulting in a sample of 25 GCs with both spectroscopy and photometry. The differences between observed and model colour-metallicity relations are consistent with the offsets observed in the two-colour diagram, with the metal-rich GCs being too red (by ≃0.2mag) in V-K and too blue (by ≃0.05mag) in V-I compared to the models at a given metallicity. These offsets cannot easily be explained as an effect of younger ages. We further compare the colour-metallicity relation for GCs in NGC 4365 with previously published data for NGC 3115 and the Sombrero galaxy, both of which are believed from spectroscopic studies to host exclusively old GC populations, and find the colour-metallicity relations for all three galaxies to be very similar. We review the available evidence for intermediate-age GCs in NGC 4365 and conclude that, while this cannot be definitively ruled out, an alternative scenario is more likely whereby all the GCs are old but the relative number of intermediate-metallicity GCs is greater than typical for giant ellipticals. The main obstacle to reaching a definitive conclusion is the lack of robust calibrations of integrated spectral and photometric properties for stellar populations with near-solar metallicity. In any case, it is puzzling that the significant intermediate-colour GC population in NGC 4365 is not accompanied by a corresponding shift of the integrated galaxy light towards bluer colours.