"As long as Simeone is here, no, I won't leave," Saul told Club del Deportista magazine.
"When a coach has faith in you as much as he has and gives you that confidence, it's vital."
Former Argentina midfielder Simeone has transformed Atletico's fortunes since taking charge of the ailing club in 2011, breaking Barcelona and Real Madrid's nine-year domestic stranglehold to win La Liga in 2014.
They also won the Europa League (2012) and the King's Cup (2013) and have also finished runner-up in two of the last three Champions League finals.
Saul is far from the only Atletico player to speak of the coach's influence, with France international Antoine Griezmann admitting in August he sought assurance Simeone would remain in charge before penning a new contract with the club.
Yet Simeone's iron grip on the club means Atletico could find themselves losing key personnel when he eventually decides to depart.
The 46-year-old recently reduced his contract with Atletico from 2020 to 2018, casting doubt on his long-term future there.
"For me and for Atletico it's the same: Simeone is everything," added Saul.
"The tranquility he transmits to me has made me the player I am today on the pitch. It took a lot to win his confidence, but now I have it I have to keep fighting to the death for him."
(Reporting by Richard Martin, editing by Pritha Sarkar)
