The 31-year-old goalkeeper gives head coach Hansi Flick another experienced option following Marc-Andre ter Stegen's loan move to Ajax earlier this month.
According to reports, Barcelona will pay two million euros for Livakovic, with a further 500,000 euros available in add-ons.
Livakovic is expected to challenge first-choice goalkeeper Joan Garcia for a starting role, while veteran Wojciech Szczesny remains available as further cover.
The Croatian spent the past two seasons at Fenerbahce, making 74 appearances and keeping 24 clean sheets. He previously enjoyed a successful spell with Dinamo Zagreb, where he won several domestic honours before moving to Turkey in 2023.
Livakovic has also been Croatia's first-choice goalkeeper for much of the past decade. He has represented his country at three World Cups and two European Championships.
His arrival makes him Barcelona's sixth signing of the transfer window, following Anthony Gordon, Karim Adeyemi, Jesse Bisiwu, Rodri and Joao Cancelo.
Barcelona began their La Liga title defence in impressive fashion with a 5-0 victory away to Elche last weekend. They now return to action against Athletic Bilbao on Thursday as they look to make another strong start to the campaign.