I may have to implement it myself.
nilearn does have for now a private function to calculate loading coefficients using Ridge Regression in the DictLearning submodule. I tried it and tweaked it, but the results are not what I expected. I need these results to correlate well with loading coefficients that I've got using ICA in Mialab's GIFT (Matlab).
For now I am using a function from GIFT to do the dual regression.
I hope I can get some free time from my boss next year to implement dual regression in nilearn (if they want that).