The Liga club have also confirmed they will not renew Jose Manuel Pinto's contract, while Rafinha and Gerard Deulofeu have returned from their loan spells
The Germany international announced in January that he would not be renewing his deal with the Bundesliga club and has long been tipped to replace Victor Valdes, who is set to leave Camp Nou this summer.
Ter Stegen's prospective move to Spain was thrown into doubt when Barcelona were handed a two-window transfer ban in April for breaching rules over the signing of international players under the age of 18, but the suspension was subsequently put on hold.
Confirmation of Ter Stegen's arrival was announced on the same day as Luis Enrique's appointment as Barcelona's new coach.
The Liga club have also revealed that Jose Manuel Pinto, who served as first-choice keeper following a season-ending knee injury to Valdes in March, will not have his contract renewed, while Rafinha and Gerard Deulofeu have returned from respective loan spells with Celta Vigo and Everton.
Deulofeu made 29 appearances for Roberto Martinez's side, scoring four goals, while attacking midfielder Rafinha, the younger brother of former Barcelona playmaker Thiago Alcantara, impressed at Celta under Luis Enrique this season, making 32 La Liga appearances.