  The polarized images of a synchrotron emitting ring are studied in the
regular Hayward and Bardeen black hole spacetimes. These regular black holes
carry a magnetic field in terms of gravity coupled to nonlinear
electrodynamics. Results show that the main features of the polarization images
of the emitting rings are similar in these two regular black hole spacetimes.
As the magnetic charge parameter increase, the polarization intensity and the
electric vector position angle in the image plane increase in Hayward and
Bardeen black hole spacetimes. Moreover, the polarization intensity and
electric vector position angle in the image of the emitting ring in the Hayward
black hole spacetime are closer to those in the Schwarzschild case. The effects
of the magnetic charge parameter on the Strokes $Q-U$ loops are also slightly
smaller in the Hayward black hole spacetime. This information stored in the
polarization images around Hayward and Bardeen black holes could help
understand regular black holes and the gravity coupled to nonlinear
electrodynamics.
