In episode 8 of season 6 there was a scene with Cersei and Qyburn after king Tommen's speech. They talked about "some rumor" and about how the birds say it's much more than just a rumor. But they don't specifically say what rumor it is. I can't remember it being mentioned in an earlier episode but I could have of course forgotten it. Is it known what rumor they were referring to?
We can only speculate until anything is shown, which most likely won't happen in S06E09; I suspect the fireworks will be reserved for the (extra-long) season finale. But
going by Qyburn's words:
"Your Grace... that old rumor you told me about."
(note the "old"),
looking at what is said next:
Qyburn: My little birds investigated.
Cersei: And? Was it just a rumor or something more?
Qyburn: More. Much more.
(with the "more" suggesting to me that he's talking about the volume of something),
and looking at the context (Cersei looking for ways to defeat the Faith Militant),
I can only think of one thing: wildfire.
There's a lot of it beneath the capital city (quoting from the Wikia):
While convalescing at Harrenhal, Jaime Lannister reveals to Brienne of Tarth that the real reason he killed Mad King Aerys II Targaryen is because he was planning on burning the entire capital city to ashes with wildfire, rather than let it fall to the rebels. When it began to appear that the rebels would win the war, the Mad King had his pyromancers hide caches of wildfire all over King's Landing - under the Great Sept of Baelor, under the slums of Flea Bottom, under houses, stables, taverns, and even under the Red Keep itself.
There's a bit of fore-warning in Jaime's words elsewhere in the episode, in his conversation with Edmure Tully:
I suppose all mothers do, but Catelyn and Cersei, there's a fierceness you don't often see. They'd do anything to protect their babies. Start a war. Burn cities to ash. Free their worst enemies. The things we do for love.
In "The Bear and the Maiden Fair", an episode from the third season (S03E07), Jaime also hinted at the disaster he averted:
The relevant part:
Qyburn: How many men have you killed, my lord?
Jaime: I don't know.
Qyburn: 50? 100? Countless.
Jaime: Countless has a nice ring to it.
Qyburn: And how many lives have you saved?
Jaime: Half a million. The population of King's Landing.
Also note Tyrion's words to Daenerys in S06E09, "Battle of the Bastards":
You once told me you knew what your father was. Did you know his plans for King's Landing when the Lannister armies were at his gates? Probably not. Well, he told my brother and Jaime told me. He had caches of wildfire hidden under the Red Keep, the Guildhalls, the Sept of Baelor, all the major thoroughfares. He would have burned every one of his citizens. The loyal ones and the traitors. Every man, woman, and child. That's why Jaime killed him.
In "The Winds of Winter", the season six finale (S06E10), we learn what rumors Qyburn and Cersei referred to:
Ser Gregor Clegane prevents King Tommen from leaving the royal palace to attend the trial of Ser Loras Tyrell and Cersei Lannister in the Great Sept of Baelor. Queen Margaery Tyrell, Lord Mace Tyrell, and Ser Kevan Lannister are already there amongst many spectators, as well as the High Sparrow.
Finally the Sparrows bring Ser Loras in. After his trial is done, Lancel tells the High Sparrow that Cersei has yet to leave the Red Keep, so the High Sparrow asks him to go retrieve her. As Lancel is walking out, one of the "little birds" lures him away into the area below the Sept. There, Lancel discovers a cache of wildfire about to be set off.
Margaery, realizing something is very wrong with the fact that neither Cersei nor Tommen are at the trial, confronts the High Sparrow. She tells him that it's a trap, but he does not listen. Margaery tells everyone in the Sept they need to leave immediately, but the Faith Militant prevent anyone from exiting. Lancel does not make it to the wildfire in time, and the Sept explodes, killing everyone inside and outside of it as well as levelling most of the surrounding area.
This recap of the events is largely based on the one available in the episode's Wikia.