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Polar vortices are omnipresent in planetary atmospheres; NASA’s Juno mission has vividly shown highly organized vortices at the poles of Jupiter. However, not much is known about their existence and characteristics in the Sun due to the present lack of direct observations at its poles. Polar vortices can strongly interact with global meridional circulation and differential rotation, leading to reversed high-latitude meridional circulation as well as polar jets in differential rotation. Does the differential rotation monotonically decrease up to the pole? Or, does the pole spin-up? Ultimately the observational discovery of polar vortices by a solar polar mission could contribute to solving mysteries of dynamical evolution of global flows and polar fields that “seed” the next solar activity cycle. Hence a better knowledge of physics of polar regions may lead to improved solar cycle forecasts and associated space weather. Unlike in planetary atmospheres, the subsurface layers of the Sun are highly influenced by the presence of magnetic fields. Here we show that solar cycle magnetic fields provide a plausible mechanism for the formation of tight vortices around the Sun’s poles. Our simulations show that the rush-to-the-poles, the migration of magnetic fields towards the pole following the Sun’s magnetic cycle, can lead to the formation of a pair of magnetohydrodynamically-dominated swirls near the pole (one cyclonic and the other, anticyclonic), which should be present in all phases of solar cycle, except during its peak-phase when the polar field reverses. By contrast, Jupiter, rotating 65 times faster than the Sun, typically has in each pole a ring of as many as 6-8 vortices, produced hydrodynamically. The mechanism proposed here for the formation of polar vortices is the first one to involve magnetic fields; it may be relevant to any star with a magnetic cycle. Our results provide observational targets for solar polar missions, which could map flows and magnetic fields near the Sun’s poles, providing important clues for understanding the origin of solar magnetism as well as its cyclic behavior.