It was Alvarez's extra-time winner against Switzerland which sent Argentina through to face England.
Asked directly about Barcelona's interest in the Atletico Madrid forward, the Barcelona president was unequivocal that his club would not be dictated to by the pace of negotiations.
"We're not going to dance to anyone's tune. Here, we set the rhythm," Laporta said. "We've made an offer, but it's not an offer without end, it's not an infinite offer.
"We'll see how long it stays on the table. We've already shown our willingness to sign the player our coach and technical management have asked for. Julian we like a lot and I think he's a great player."
Laporta also addressed the state of Barcelona's relationship with Atletico Madrid, which had appeared strained after a public tug-of-war over the striker.
"I understand we have a very good relationship with them. There was some confusion regarding the offer we made and I clarified it with them. We haven't pushed any further. I only said that once they have an alternative, our offer would still stand. And it stayed there. It hasn't moved beyond that, for now," he said.
The Barcelona president is in Texas to support the club's players on international duty, and struck a sympathetic note for one absent face.
"It's the moment to give support, we have a lot of players in Spain who will be very happy if they reach the final. I feel for Jules Kounde, who's in France, but... we have good representation," Laporta said, before praising Lamine Yamal's tournament. "He's having a great World Cup and has already been named MVP a couple of times. But not just him: Pedri, Olmo and Cubarsi too."
On Ferran Torres, whose future has been the subject of growing speculation, Laporta declined to comment beyond praising his form in the tournament. He was more forthcoming on Raphinha, ruling out any prospect of a sale.
"We have no interest in Raphinha leaving Barca. Quite the opposite, he's a cornerstone player for our team. He's key for us. The shame is he couldn't be at full fitness during the decisive stretch of the season," he said.
Laporta's comments on Alvarez add another layer to a saga that has dragged on for weeks, with Atletico publicly resistant to strengthening a domestic rival and both Barcelona and Real Madrid accused by the Madrid club of campaigning too openly for the striker.
The Barcelona president's insistence that the offer remains on the table, but not indefinitely, suggests the club is keen to force a resolution one way or another rather than let the situation drift deeper into pre-season, with Alvarez himself continuing to focus on Argentina's push for a second consecutive World Cup title in the meantime.