I made a theory saying that Minato had sage mode but I got flamed.
During Naruto's training, the frog said only two people have mastered this frog sage mode, and when Naruto got to master the sage mode the frog said to Gamabunta that he reminds him of those two. Also, after he took out Pain with one blow he repeated the statement again but this time the picture of Jiraya boy and Minato were shown and the frog said that Naruto has finally surpassed them.
Also, if you monitor and read the manga carefully… you will notice that Minato was in the Kurama Chakra mode from when he saw that Naruto had gotten Kurama under his control. He didn't know/didn't have the ability to combine both Sage mode and Kurama mode. Even Naruto didn't know as well until Kurama told him that it happened in his battle against Pain.
However, everyone should have known that Minato had sage mode because he was able to hold a rasengan that had Natural energy chakra, when he told Naruto to add the natural chakra to his rasengan… Just like sasuke was able to add Juugo's natural chakra to susanoo because he is compatible with it. The only thing different is that Minato was unable to use sage mode in Kurama chakra mode.