The Barcelona president made clear the club's patience with Atlético Madrid over the striker will not be unlimited.
"There is a very important offer on the table for Julián," Laporta said. "Atlético already know we've made an important offer. Very important. If they're willing to accept it, fantastic.
"What it won't be is open-ended. There will come a point where we have to decide whether we keep the offer on the table or not, and that will depend on how these two weeks go before the end of July. We've done this with institutional respect."
Reports in the Spanish press have put Barcelona's improved proposal at around €130m, a sharp rise on the €100m opening bid the club tabled with Atlético back in May, which the Rojiblancos dismissed at the time as unrealistic and insisted their star forward was not for sale.
Álvarez has been strongly linked with a move to Camp Nou as a long-term replacement for Robert Lewandowski, who left for Chicago Fire this summer, with the Argentina international having grown into one of Atlético's most important players since his move from Manchester City in 2024.
Laporta was cautious when pressed on the finer details of the negotiation, choosing his words carefully in front of the Spanish media.
"I don't want to talk about it. The player has spoken and the matter is live. We'll see what Atlético say. It's very clear the player wants a change of scenery, and we're in a position to welcome him and, what's more, making a very good offer. The offer is very good. There it is. We'll see how this develops," he said.
The Barcelona president also gave an update on the club's wider transfer business.
"Deco is working very well. This week we signed Adeyemi. He's working very efficiently and with discretion. We'll see how things develop with Julián — if it's not certain, there are alternatives," Laporta added.
Álvarez himself fuelled the speculation during the World Cup, telling reporters after Argentina's group-stage win over Austria that a transfer would suit all parties and that he wanted to fulfil his dream of joining Barcelona.
Atletico Madrid, for their part, have publicly insisted the forward is not for sale, though Laporta's latest comments suggest Barcelona believe the deal is edging closer to completion as the transfer window enters its decisive final weeks.