The 22-year-old, who impressed as Spain’s Under-21 side lifted the European Championship in June, was Barca’s top target as the Catalans sought a back-up option for Jordi Alba this summer.
Firpo had a release clause of €50 million and Betis were thought to want €30m initially for the Dominican-born left-back.
But he looks poised to head to the Camp Nou after two clubs agreed a fee of €18m plus €7m in add-ons.
Barca considered a number of options this summer for the left-back position, including Bayern Munich’s David Alaba, Borussia Dortmund’s Raphael Guerreiro and former Atletico Madrid defender Filpe Luis, who recently signed for Flamengo in Brazil.
Coach Ernesto Valverde wanted the signing completed before the team head off to the United States on Monday and Firpo should join his team-mates for the trip, which features fixtures against Napoli in Miami and Michigan.
Junior played 29 games for Betis last season, scoring three goals, including a fine effort in the 4-3 win over Barcelona at Camp Nou in November.