The 43-year-old Terzic will replace Ernesto Valverde, who said in March that he would leave Bilbao at the end of the current campaign after four years in charge.
Terzic was previously in charge of Borussia Dortmund, winning the German Cup in 2021 and narrowly missing out on the Bundesliga title in 2023.
He also led Dortmund to the UEFA Champions League final in 2024 but lost to Real Madrid at Wembley before leaving and has not been in management since then.
Terzic had also been Slaven Bilic's assistant at Turkish club Besiktas and English Premier League side West Ham.
The departing Valverde, who has had three tenures as Bilbao coach, ended their 40-year trophy drought when his side won the Copa del Rey in 2024.
Bilbao are currently eighth in Spain's La Liga standings with four games of the season left with Valverde looking for a strong finish to the campaign.
