Joan Laporta has set a deadline of the end of July or start of August for a definitive answer from Atlético, having confirmed the Catalan club's offer for Álvarez stands at around €100m.
"We've made a very important offer. If they're willing to accept it, fantastic, but it won't be open-ended. We have to make a decision," Laporta said of the proposal, which Atletico Madrid chief executive Miguel Ángel Gil Marín has repeatedly insisted the club will not accept even if it were doubled to €200m.
Real Madrid also tabled an offer worth a reported €150m for the World Cup winner earlier this month, which Atlético rejected just as firmly, with Diego Simeone's side expecting the Argentine to report for pre-season training on 10 August regardless of his stated desire to leave for Barcelona, a wish he first communicated to the club's hierarchy in May and is expected to repeat in the coming days.
With Atlético's position showing no sign of softening, Barcelona's sporting department is exploring alternatives rather than waiting indefinitely.
Bournemouth forward Eli Junior Kroupi, 20, is seen as the other leading candidate, but the Cherries consider him untransferable and have set a price of around €130m, a sum Barcelona are not prepared to pay despite regarding the France Under-21 international as a strong long-term investment; Tottenham, Arsenal and Paris Saint-Germain are among the other clubs also tracking him.
Chelsea's Joao Pedro, 24, has likewise been sounded out by Barcelona executives, but his exit is considered close to impossible given he is contracted to Stamford Bridge until June 2033.
Barcelona are not expected to break the bank for any of the alternatives, not least because the squad already has options capable of covering Robert Lewandowski's departure, including Ferran Torres, Dani Olmo, Lamine Yamal and Anthony Gordon.
Sporting director Deco has other names on his list of potential signings and has opened preliminary talks with some of them, though nothing is close to being finalised, and he remains ready to move for a replacement should any current forward, such as Ferran Torres, leave the club this summer. For now, though, Álvarez remains the priority, with the club hopeful that pressure from the player himself will eventually force a change in Atlético's position.