On Earth cyclones have regions of calm weather commonly known as eyes. They appear to create tunnels spreading from the top of the atmosphere down to the sea level. Wikipedia article mentions their presence in other planets as well.
NASA reported in November 2006 that the Cassini spacecraft observed a "hurricane-like" storm locked to the south pole of Saturn with a clearly defined eyewall. The observation was particularly notable as eyewall clouds had not previously been seen on any planet other than Earth (including a failure to observe an eyewall in the Great Red Spot of Jupiter by the Galileo spacecraft). In 2007, very large vortices on both poles of Venus were observed by the Venus Express mission of the European Space Agency to have a dipole eye structure.
Atmosphere probes don't seem to be able to reach very deep. Contact with Galileo probe ended after around 160 km and Saturn Atmospheric Probe is expected to survive up to 250 km. Could we reach even deeper if we travel through an eye of a gas giant vortex? Is targeting these viable?