I'd like to keep him as well, but I think this may be a case of big Nat pushing for a move himself. In a way, he probably feels a bit let down with how we've treated him.
We kept him from that move to the US a few years ago possibly with the promise of a pathway to the first team that never came. Even last year, despite injuries completely wrecking us, he didn't get a chance until after about 6 months.
Klopp clearly doesn't like slow defenders in that high line. Would you really blame him for suspecting that he'll be frozen out again if van Djik, Gomez, Konate and Matip are all fit? Of course all 4 being fit seems unlikely to us, but is Phillips really going to stake his career on the hope of injuries?
He probably thinks this is the best time to move while his stock is high, and he can get offers from a good mid-table side.