Others already replied but LF definitely knows.
He was in love with Catelyn and made it his business to know everything about her and her family. He basically told Sansa in the crypt that the official story as told by Robert and the victors of the war might not be true, hinting that she wasn't kidnaped but rather ran away to be with the Prince.
My guess is he was behind the assassin to kill Bran in season 1 (after his fall) because he wanted the Stars and Lannisters at war (to get rid of Ned also). He was already behind killing John Arryn and knew blame would fall on Jamie and Cersei once their incest was discovered.
It figures he knows this vital piece of information and will possibly use it against Jon and top possibly elevate Sansa as the true Stark who should be leading Winterfell.
Varys had birdies everywhere and like LF, makes it his business to be knowledgeable about all things happening in the realm. It's how both of these characters who are nobodies by name have managed to survive and thrive.
On another topic, any theories or ideas on what the Red Witch meant in her last convo with Varys about both of them having to return to die on this strange land? I'll miss her and her exuberance to get nekkid at a drop of the hat