Is Tsunade's son Minato?
He is the reincarnation of Asura but that isn't enough proof. So, now let's take a look at his lineage. Naruto was born to Minato Namikaze and Kushina Uzumaki. ... So, that brings us to the theory which is,
Minato is Tsunade and Dan's son.
Did Tsunade reject Jiraiya?
A princess of the village tried to fight Jiraiya for his hand in marriage,
but he declined. He told her that his heart belonged to another. Then he described the hardships that Tsunade has gone through, and the fact that, despite his love being one-sided, he is devoted to making her life happier.