We present and investigate a novel approach towards broad-bandwidth near-field scanning optical spectroscopy based on an in-line interferometer for homodyne mixing of the near field and a reference field. In scattering-type scanning near-field optical spectroscopy, the near-field signal is usually obscured by a large amount of unwanted background scattering from the probe shaft and the sample. Here we increase the light reflected from the sample by a semi-transparent gold layer and use it as a broad-bandwidth, phase-stable reference field to amplify the near-field signal in the visible and near-infrared spectral range. We experimentally demonstrate that this efficiently suppresses the unwanted background signal in monochromatic near-field measurements. For rapid acquisition of complete broad-bandwidth spectra we employ a monochromator and a fast line camera. Using this fast acquisition of spectra and the in-line interferometer we demonstrate the measurement of pure near-field spectra. The experimental observations are quantitatively explained by analytical expressions for the measured optical signals, based on Fourier decomposition of background and near field. The theoretical model and in-line interferometer together form an important step towards broad-bandwidth near-field scanning optical spectroscopy.