Though the audience has been let in on the secrets of Jon Snow's heritage (namely that he's the child of Lyanna Stark and Rhaegar Targaryen), there aren't many characters on Game of Thrones who could say the same. In fact, there's only one who's in the right place at very nearly the right time: Jon's "brother" (his actual cousin) Bran Stark, who witnessed Jon's birth in a psychic vision in Season 6. Bran was missing and assumed to be dead until he arrived back at Winterfell with some brand new powers and a lot less tact. He has all the information, but will Bran reveal Jon Snow is a Targaryen on Game of Thrones?
That definitely seems to be Bran's goal, but circumstances may prevent him from accomplishing it any time soon. During his tepid reunion with Sansa, Bran mentioned wanting to talk to Jon, though he didn't share whether or not he was going to spill the beans when he did. With Jon currently cooling his heels at Dragonstone while he tries to make a deal with Daenerys, there's no telling when Bran and Jon will meet up again. Bran could always send a raven Jon's way, but that's like breaking up with someone via text. You don't tell your brother he's not your brother in a letter.
Actor Isaac Hempstead Wright mentioned the difficulty of disclosing such delicate information in an interview with The Hollywood Reporter. "But I don't know how Bran's going to break it to Jon," he said. "He hasn't seen him in years. 'Hey, yeah, but the way, I'm a tree wizard now and Dad's not your father.' I think Jon will go, 'You've been spending way too much time in the cave, mate.'"
Considering what Jon's lived through, it might be pretty easy to convince him to believe the unbelievable. But when it comes to Bran revealing the truth to Jon, the main problem is the distance between them; that's always a problem for the Starks. Bran is the only person capable of telling Jon where he came from, but unfortunately he might not get the opportunity any time soon. Until Jon arrives back at Winterfell, he's out of the loop.
The dark family secrets may have to wait until the Starks (or Stark-Targaryens, as the case may be) are all assembled in their home once again. It took six years for that to happen in the first place, but hopefully it doesn't take that long again.