We use a scanning nanometer-scale superconducting quantum interference device to map the stray magnetic field produced by individual ferromagnetic nanotubes (FNTs) as a function of applied magnetic field. The images are taken as each FNT is led through magnetic reversal and are compared with micromagnetic simulations, which correspond to specific magnetization configurations. In magnetic fields applied perpendicular to the FNT long axis, their magnetization appears to reverse through vortex states, that is, configurations with vortex end domains or in the case of a sufficiently short FNT with a single global vortex. Geometrical imperfections in the samples and the resulting distortion of idealized magnetization configurations influence the measured stray-field patterns.