Not really. For an infinitesimal loop, there is only magnetic field up close to the loop. The loop is, to first order, an equipotential, so there is no electric field. The electric field 'appears' as the wave propagates away from the antenna, by the fluctuation of the magnetic field.
It is the dual of the case for an infinitesimal electric dipole, which only has electric field adjacent to it, where the magnetic field appears as a result of the propagating electric field. In the electric dipole, there is no length (to first order) that the current flows through, to generate a magnetic field.
For finite sized loops and dipoles, there are indeed electric and magnetic fields, as the loop is not equipotential, and currents flow in the electric dipole. For small loops or dipoles, one field is much stronger than the other (in the near field). The near field falls of as the cube of the distance, so that after a few wavelengths, only the propagating far field remains.
When the loop or dipole gets to resonant size, see the 'classic' antennae in various text books, it is rather easier to spot how both fields are generated.