The production of φ mesons in the reaction e+p → e+φp (φ → K+K-), for 7 < Q2 < 25 GeV2 and for virtual photon-proton centre of mass energies (W) in the range 42-134 GeV, has been studied with the ZEUS detector at HERA. When compared to lower energy data at similar Q2, the results show that the γp → p0p cross section rises strongly with W. This behaviour is similar to that previously found for the γ*p → p0p cross section. This strong dependence cannot be explained by production through soft pomeron exchange. It is, however, consistent with perturbative QCD expectations, where it reflects the rise of the gluon momentum density in the proton at small x. The ratio of σ(φ)/σ(p0), which has previously been determined by ZEUS to be 0.065 ± 0.013 (stat.) in photoproduction at a mean W of 70 GeV, is measured to be 0.18 ± 0.05 (stat.) ± 0.03 (syst.) at a mean Q2 of 12.3 GeV2 and mean W of ≈ 100 GeV and is thus approaching at large Q2 the value of 2/9 predicted from the quark charges of the vector mesons and a flavour independent production mechanism.