Simpson-Visser (SV) spacetimes are the simplest globally regular modifications of the Schwarzschild, Reissner-Nordsröm and other black hole solutions of general relativity. They smoothly interpolate between these black holes and traversable wormholes. After a brief presentation of the Schwarzschild-like and Reissner-Nordsröm-like SV geometries, including their Carter-Penrose diagrams, we show that any static, spherically symmetric SV metric can be obtained as an exact solution to the Einstein field equations sourced by a combination of a minimally coupled phantom scalar field with a nonzero potential V(ϕ) and a magnetic field in the framework of nonlinear electrodynamics with the Lagrangian L(F),
F=FμνFμν (in standard notations). Explicit forms of V(ϕ) and L(F) are presented for the cases of Schwarzschild-like and Reissner-Nordsröm-like SV metrics.