Firpo, 24, is also wanted by Southampton and several Italian sides, but Leeds have stepped up negotiations with Barca over the last few days and the left-back is keen to make the switch to Elland Road.
Barca and Leeds are still in talks regarding the structure of the deal. They need to agree on a price and on whether the English side sign Junior outright or on loan with a clause to make the move permanent next summer.
Leeds' director of football Victor Orta has played a key role in selling the club to Junior, who wants to leave Barca in search of regular action and feels he would fit well in Marcelo Bielsa's system.
However, Barca prefer a permanent move to a loan and the English clubs are in the best position to pay a fee.
Junior joined Barca for an initial €18 million from Real Betis in 2019 with the final cost rising to as much as €30m based on a series of add-ons.
Sources have told ESPN that the Catalan club want to pay off the amortisation on that transfer, which is somewhere between €10.8m and €18m depending on how many add-ons have been met.
In two seasons at Camp Nou, Junior, a former Spain Under-21 international, has made 41 appearances for the first-team.
Barca, whose gross debt is over €1 billion, need to bring in money this summer to help fund arrivals and also need to reduce the wage bill to be able to register new players within the league-imposed salary cap.