The 17-year-old was a key player for Barcelona during the 2021-22 campaign, making 46 appearances in all competitions, contributing two goals and six assists in the process.
In La Liga, he featured on 34 occasions, contributing two goals and six assists, while he played 11 times in the Champions League and Europa League during a breakthrough season.
Gavi's contract at Camp Nou is due to expire next June, though, and there has been a host of speculation surrounding his future in recent months.
However, Laporta has suggested that Barcelona are now reaching the latter stages of their negotiations with Gavi's agent, and an announcement on a new deal could be imminent.
"Gavi is doing very well. We are very satisfied with how everything has developed with Ivan de la Pena and with the player," Mundo Deportivo quotes Laporta as saying.
The midfielder became Spain's youngest ever player when he debuted last October, and he is in a strong position to make the La Roja squad for the 2022 World Cup.
"We're optimistic because we want Gavi to stay at Barcelona. We are very excited about him staying, he's from our academy," Laporta told Marca when questioned on the situation last month.
"The player and his agent, who we know, also want to stay at Barcelona. We are very close to reaching an agreement."
However, the negotiations have not been smooth, with Laporta admitting in May that the club were becoming frustrated with the lengthy discussions.
"His agent has had the renewal offer on the table for a long time," Laporta told L'Esportiu. "We don't have any news they have accepted it. The news that we do have is that they are comparing and at some time will say something. We have said our position and for now, it's not been accepted.
"Yes (hurt by the actions of Gavi's agent), because we don't understand it. Gavi is a player that we all like, he has a present and a magnificent future at Barcelona.
"Within the salary levels, it's more than acceptable. (But) we will not go out of these levels, because I don't want Barca to continue in the line that those before us were on that has taken Barca to ruin.
"If we have players that come out of the academy, we make them an offer of a present and a very good future, and if the agent is comparing and delaying the decision, it's understandable that I don't like the situation."