Using an optimally coupled nanometer-scale SQUID, we measure the magnetic flux originating from an individual ferromagnetic Ni nanotube attached to a Si cantilever. At the same time, we detect the nanotube's volume magnetization using torque magnetometry. We observe both the predicted reversible and irreversible reversal processes. A detailed comparison with micromagnetic simulations suggests that vortexlike states are formed in different segments of the individual nanotube. Such stray-field free states are interesting for memory applications and noninvasive sensing.