Gavi is convinced Real Madrid are "worried" about the LaLiga champions.
He told AS: "Are Real Madrid worried? Of course they are.
"We've won four out of four El Clasicos, and that's never happened before. It's logical that they're worried. Ultimately, we were spectacular last season, and they didn't win anything."
For the first time since 2021, Real Madrid finished last season without winning a major trophy.
This summer, Real Madrid have brought in the likes of Trent Alexander-Arnold, Dean Huijsen, Alvaro Carreras and Frank Mastantuono.
"They've gotten stronger this year, with some really good players, to be honest," Gavi said. "But we'll see what they do next season. They have a fantastic team, but we just have to be ourselves."
On Barca's summer market, Gavi also said: "I am convinced that (Joan) Garcia and (Marcus) Rashford will make a great contribution. (Roony) Bardghji is also doing very well. I hope they give their all to win all the titles. I am happy with the arrival of Marcus because he is an impressive player."
And on Barca's new No10, Lamine Yamal, he added: "He's an incredible player. He's improving every day. He's extraordinary in training. He has to continue like this, with this mentality, and I'm sure that if he doesn't win the Ballon d'Or this year, he'll win many in the future."