  We show that Abelian Higgs Models with dielectric function defined on the
noncommutative plane enjoy self-dual vorticial solutions. By choosing a
particular form of the dielectric function, we provide a family of solutions
whose Higgs and magnetic fields interpolate between the profiles of the
noncommutative Nielsen-Olesen and Chern-Simons vortices. This is done both for
the usual $U(1)$ model and for the $SU(2)\times U(1)$ semilocal model with a
doublet of complex scalar fields. The variety of known noncommutative self-dual
vortices which display a regular behaviour when the noncommutativity parameter
tends to zero results in this way considerably enlarged.
