In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra $\mathfrak{osp}(1|2)$ in terms of a so-called radially deformed Dirac operator $D$ depending on a deformation parameter $c$ such that for $c=0$ the classical Dirac operator is reobtained. In this paper, several versions of the Paley-Wiener theorems for this radially deformed Fourier transform are investigated, which characterize the supports of functions associated to this generalized Fourier transform in Clifford analysis.