Sky Sports is reporting that Smith, 22, will now negotiate personal terms with Bournemouth, who finished 16th last season, after Liverpool accepted a bid.
Smith played 10 times for Liverpool last season, and while he has been praised by manager Jurgen Klopp on numerous occasions, he is considered second choice behind Alberto Moreno.
The Liverpool Echo is reporting that the move is worth £6million ($10.54M) and comes off the back of Bournemouth purchasing Jordan Ibe from Liverpool for £15m ($26.35m).