I think the rumor from Bild was 20 million, but that was probably in Euros, and Bild puts almost anything out there -- Kicker seems to be more reliable. So, I don't doubt that Stoke has interest, but I doubt the valuations are that far apart.
That being said, all parties have to agree, so if Pulisic is set on moving but really wants Liverpool, they can't get him for whatever price they want, but they might be able to get him for less. That's also not considering add-ons. If the Liverpool press is reporting 11 million pounds, then I wouldn't be surprised if that was on the low end of the potential value of the transfer. 11 million pounds + for a 17 year old is pretty good cash. Klopp brought Pulisic through to the first team to train when he was 16. He didn't get a full run out until Tuchel, but apparently Klopp loves Pulisic, and Pulisic and his dad love Klopp.
I think he should stay, but Dortmund also just bought Emre Mor, Dembele, Schürrle, and Götze, for relatively big money, to play in the positions he'd most likely see time -- that's in addition to Kagawa and Reus, none of whom are really old, so you can see where Pulisic might want to keep his options open; plus, Liverpool would probably pay him more.