This paper concerns a relationship between the kernel of the Bargmann transform and the corresponding canonical transformation. We study this fact for a Bargmann transform introduced by Thomas and Wassell [J. Math. Phys. 36, 5480–5505 (1995)]—when the configuration space is the two-sphere S2 and for a Bargmann transform that we introduce for the three-sphere S3. It is shown that the kernel of the Bargmann transform is a power series in a function which is a generating function of the corresponding canonical transformation (a classical analog of the Bargmann transform). We show in each case that our canonical transformation is a composition of two other canonical transformations involving the complex null quadric in C3 or C4. We also describe quantizations of those two other canonical transformations by dealing with spaces of holomorphic functions on the aforementioned null quadrics. Some of these quantizations have been studied by Bargmann and Todorov [J. Math. Phys. 18, 1141–1148 (1977)] and the other quantizations are related to the work of Guillemin [Integ. Eq. Operator Theory 7, 145–205 (1984)]. Since suitable infinite linear combinations of powers of the generating functions are coherent states for L2(S2) or L2(S3), we show finally that the studied Bargmann transforms are actually coherent states transforms.