Human rhythmic movements spontaneously entrain to external rhythmic stimuli. Such sensory-motor entrainment can attract movements to different tempi and enhance their efficiency, with potential clinical applications for motor rehabilitation. Here we investigate whether entrainment of self-paced rhythmic movements can be induced via transcranial alternating current stimulation (tACS), which uses alternating currents to entrain spontaneous brain oscillations at specific frequencies. Participants swung a handheld pendulum at their preferred tempo with the right hand while tACS was applied over their left or right primary motor cortex at frequencies equal to their preferred tempo (Experiment 1) or in the alpha (10Hz) and beta (20Hz) ranges (Experiment 2). Given that entrainment generally occurs only if the frequency difference between two rhythms is small, stimulations were delivered at frequencies equal to participants' preferred movement tempo (≈1Hz) and ±12.5% in Experiment 1, and at 10Hz and 20Hz, and ±12.5% in Experiment 2. The comparison of participants' movement frequency, amplitude, variability, and phase synchrony with and without tACS failed to reveal entrainment or movement modifications across the two experiments. However, significant differences in stimulation-related side effects reported by participants were found between the two experiments, with phosphenes and burning sensations principally occurring in Experiment 2, and metallic tastes reported marginally more often in Experiment 1. Although other stimulation protocols may be effective, our results suggest that rhythmic movements such as pendulum swinging or locomotion that are low in goal-directedness and/or strongly driven by peripheral and mechanical constraints may not be susceptible to modulation by tACS.