James Trafford helped Bolton reach the League One play-offs and win the Papa Johns Trophy on loan last season
City have a 20% sell-on clause and a buy-back option within the deal, which is £15m up front and £4m add-ons.
Trafford, 20, is yet to appear in the Premier League and has spent the previous two seasons on loan in League One with Accrington and Bolton.
Burnley face champions City in their top-flight opener on 11 August.
The £19m deal will make Trafford Burnley's record signing, eclipsing the £15m paid for Chris Wood and Ben Gibson in 2017 and 2018 respectively.
He becomes Burnley's second major signing in as many days after the club confirmed the arrival of Switzerland forward Zeki Amdouni from Basel for an undisclosed fee on Wednesday.
Trafford kept a record six clean sheets as England won the European Under-21 Championship without conceding a goal earlier this month.
When Spain won a penalty deep into injury time in the final, Trafford brilliantly saved Abel Ruiz's before keeping out a rebound.
He is the second goalkeeper to arrive at Turf Moor this summer.
Manager Vincent Kompany signed goalkeeper Lawrence Vigouroux from Leyton Orient while another former City player, Kosovo international Arijanet Muric, was the club's number one last term, featuring 41 times as they won the Championship convincingly.
Burnley also signed Republic of Ireland defender Dara O'Shea this summer, while Jordan Beyer's loan move from Borussia Monchengladbach was made permanent in May.
-- Courtesy of BBC Sport