One of the most formative things that ever happened to poor Lord Varys was his traumatic castration at the hands of a magician from Myr. As he told Tyrion back in Season 3:
I still dream of that night. Not of the sorcerer, not of his blade—I dream of the voice from the flames. Was it a god, a demon, a conjurer’s trick? I don’t know. But the sorcerer called and a voice answered and ever since that day I have hated magic and all those who practice it.
Up until now, we didn’t know what the voice in the flame said to Varys. But we know the followers of R’hllor somehow do. Kinvara, the high-ranking red priestess who paid Tyrion and Varys a visit in Meereen last season, made it clear that the Spider’s voice-in-the-flames moment was known to her.
Do you remember what you heard that night when the sorcerer tossed your parts in the fire? You heard a voice call out from the flames, do you remember? Should I tell you what the voice said? Should I tell you the name of the one who spoke?
It stands to reason that Melisandre knows about this too, and the look of recognition Varys gives her as she drops a comment about their shared appointment with death could indicate that the Spider heard his own demise foretold by R’hllor (or whatever you want to call it) in the fire as a child.