In this paper we use solar neutrino data to derive stringent bounds on Majorana neutrino transition moments (TMs). Should such be present, they would contribute to the neutrino--electron scattering cross section and hence alter the signal observed in Super-Kamiokande. Motivated by the growing robustness of the LMA-MSW solution of the solar neutrino problem indicated by recent data, and also by the prospects of its possible confirmation at KamLAND, we assume the validity of this solution, and we constrain neutrino TMs by using the latest global solar neutrino data. We find that all elements of the TM matrix can be bounded at the same time. Furthermore, we show how reactor data play a complementary role to the solar neutrino data, and use the combination of both data sets to improve the current bounds. Performing a simultaneous fit of LMA-MSW oscillation parameters and TMs we find that 6.3 times 10^{-10} mu_B and 2.0 times 10^{-10} mu_B are the 90% C.L. bounds from solar and combined solar + reactor data, respectively. Finally, we perform a simulation of the upcoming Borexino experiment and show that it will improve the bounds from today's data by roughly one order of magnitude.