The 17-year-old's current contract is due to expire next June, and there has been a host of speculation surrounding his future in recent months, with Premier League clubs said to be interested in his signature.
However, there have been a series of meetings between Barcelona and Gavi's representatives in recent weeks with the view to coming to an agreement over a new long-term deal.
According to journalist Helana Condis Edo, all parties are now 'getting closer' following a meeting between the club and the player's agent Ivan de la Pena.
Italian journalist Fabrizio Romano claims that the length of the contract is yet to be agreed, but there will be a release clause of £860m in the proposed new deal.
Gavi was a vital player for Barcelona during the 2021-22 campaign, making 46 appearances in all competitions, contributing two goals and six assists in the process.
The midfielder became Spain's youngest ever player when he debuted last October, while he ended the 2021-22 La Liga campaign with two goals and six assists in 34 appearances.
Earlier this month, Barcelona president Joan Laporta told Marca: "We're optimistic because we want Gavi to stay at Barcelona. We are very excited about him staying, he's from our academy.
"The player and his agent, who we know, also want to stay at Barcelona. We are very close to reaching an agreement."
However, there have been difficulties in the negotiations, with Laporta admitting last month that the club were becoming increasingly frustrated by the situation.
"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."
As well as playing a key role in Barcelona's 2021-22 La Liga season, the midfielder also featured six times in the Champions League and five times in the Europa League last term.