Tailoring the light-matter interaction and the local density of optical states (LDOS) with nanophotonics provides accurate control over the luminescence properties of a single quantum emitter. This paradigm is also highly attractive to enhance the near-field Förster resonance energy transfer (FRET) between two fluorescent emitters. Despite the wide applications of FRET in nanosciences, using nanophotonics to enhance FRET has remained a debated and complex challenge. Here we demonstrate enhanced energy transfer within single donor-acceptor fluorophore pairs confined in single gold nanoapertures. Experiments monitoring both the donor and the acceptor emission photodynamics clearly establish a linear dependence of the FRET rate on the LDOS in nanoapertures, demonstrating that nanophotonics can be used to intensify the near-field energy transfer. Strikingly, we observe a significant six-fold increase in the FRET rate for large donor-acceptor separations exceeding 13 nm. Exciting opportunities are opened to investigate biochemical structures with donor-acceptor distances much beyond the classical Förster radius. Importantly, our approach is fully compatible with the detection of single biomolecules freely diffusing in water solution under physiological conditions.