The Blues will pay and initial £53 million with a further £5m in performance-related add-ons.
Personal terms are not expected to be an issue and the 19-year-old will undergo a medical later this week ahead of a move to Stamford Bridge.
Liverpool had previously had a £60m ($76.2m) offer for Lavia accepted by Southampton but the player expressed a desire to join Chelsea, allowing the Blues to negotiate a slightly lower fee.
It is the second time in a matter of days that Liverpool have lost out on a midfielder to Chelsea after Jurgen Klopp's side agreed a British record £111m fee with Brighton for Moisés Caicedo, only for the player to reject the proposal.
Chelsea signed Caicedo on Monday in a deal worth up to £115m ($146m).
Saints paid an initial £10.5m but are due to pay a 20 per cent sell-on fee to City after agreeing terms with Chelsea.