The 21-year-old's future had been subject of much speculation following an impressive two seasons in the Atletico first team, but the midfielder signed a new five-year contract at the Vicente Calderon earlier this week.
"This is the team where I grew up, and it's the best place for me to continue growing. I like to win and now I'm winning with Atleti. I'm very happy to continue fighting for this shirt. It's a dream to stay here," Saul told Marca.
Saul, whose buy-out clause has risen to just over £60m, has been included in the provisional Spain squad for the European Championships this summer.