Bale's future in Spain had been unclear, with the Welshman admitting after the 2018 Champions League final that he could leave Los Blancos ahead of the 2018-19 campaign.
Cristiano Ronaldo's exit to Juventus appears to have strengthened Bale's position at Madrid, however, and despite reported interest from Manchester United, Lopetegui has backed the attacker to star in the Spanish capital next term.
"He's very, very happy. He's very happy to play at Real Madrid. He's happy to be here, this is a fantastic opportunity for him to show his talent," Lopetegui told reporters.
"I've talked to him like I've talked to everyone. He's with the team, he's happy, he's in line with our goals and it's a start of a new exciting path for us that I'm looking forward to.
"We first recognise Cristiano as one of the most important players in the recent history of Real Madrid, he vocalised his desire to leave and the club gave him all the facilities to do so. We wish him all the luck in the world.
"And when it comes to our players, Bale and everyone else, we are completely convinced we have the players to move forward. Bale is a magnificent player with qualities and we are convinced he and everyone else can fill the void."
Lopetegui's team will begin their preparations for the new campaign with a pre-season friendly against Man United on Tuesday night.