Due to evidence in the manga it is my belief that Jiraiya is Minato's father. Here are some reason's why:
1. They have very similar and almost identical traits. These are identical hair ( except for jiraiya's ponytail ), same blood type ( B ), believable age difference, similar personalities ( shrewd, sense of humor, except jiraiya's perverted behavior), and both have large chakra reserves and great control despite coming from unknown clans/ families.
Evidence:
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2. You have my evidence that made me believe it now its time I tell you how it could actually happen or play out in the story.
• Jiraiya is from an unknown yet powerful clan. He loves Tsunade yet she rejects him. He fights for her love to no avail. Finally in a desperate attempt later in life ( around his 30's ) at revenge or to get Tsunade jealous he gets with Tsunade' cousin, who is actually Tobirama's Granddaughter.Tobirama had a daughter who married a Namikaze man who was also militant like Tobirama. After Jiraiya got with Tsunade's cousin she got pregnant, little did Jiraiya know at that time it was his child. She was a bit of a fluezy as an act of rebellion against her militant and controlling father. Thus she never knew who the father was and the baby Minato took his mother's last name. Namikaze.
• Once Jiraiya took Minato under his wing he noticed the shocking resemblance yet blew it off as a coincidence. Near this time Tsunade would come to realize that Minato is Jiraiya's child but doesn't tell him. This is why despite developing feelings for him she never acts on them. One scene to remember is when they talk about Minato before Jiraiya leaves to battle Pain. In this scene Jiraiys says he doesn't have any children but if he did Minato would be a good one or something along those lines. Tsunade gives a sullen look and gazes off in remembrance. This is ironic because if Minato Is Jiraiya's son then he would have been a proud parent of he knew yet the sullen look Tsunade gives is because she knows Minato is his son and she's debating telling him but doesn't because it would probably hurt him. She may have also felt bad she never got with him which is why she said if he comes back she will. Spoiler alert. He doesn't.
My theory on how it will play out from here is where Tsunade is alive and heading to the battlefield where Minato is she will explain that Jiraiya is Minato's father andit will motivate Minato and Naruto to defeat the juubi and madara to ensure Jiraiya didn't die in vain. It would also work out perfect if Jiraiya is revived and finds out. This would bring everything full circle and explain how naruto has senju DNA to illustrate the connection between naruto and sasuke and the senju-uchiha rivalry.
D