In this paper, we investigate the oscillation of a fullerene that is moving within the centre of a bundle of nanotubes. In particular, certain fullerene–nanotube bundle oscillators, namely C60-carbon nanotube bundle, C60-boron nitride nanotube bundle, B36N36-carbon nanotube bundle and B36N36-boron nitride nanotube bundle are studied using the Lennard–Jones potential and the continuum approach which assumes a uniform distribution of atoms on the surface of each molecule. We address issues regarding the maximal suction energies of the fullerenes which lead to the generation of the maximum oscillation frequency. Since bundles are also found to comprise double-walled nanotubes, this paper also examines the oscillation of a fullerene inside a double-walled nanotube bundle. Our results show that the frequencies obtained for the oscillation within double-walled nanotube bundles are slightly higher compared to those of single-walled nanotube bundle oscillators. Our primary purpose here is to extend a number of established results for carbon to the boron nitride nanostructures.