The 23-year-old spent time in Real Madrid's youth system between 2006 and 2008 before moving to Atletico.
The versatile midfielder has made over 200 appearances for Atletico's first team since breaking through and is believed to be on the radar of a number of European clubs.
United were said to be prepared to pay £134m to sign the Spaniard, but Saul has said that he signed a new long-term contract at Atletico, which runs until 2026, with the view to retiring at the club.
"That's why I signed a long-term contract. That's my intention, to be at Atletico Madrid all my life, but in the world of football you never know," Saul told Radio Nacional Espana. "For me, yes [I want to retire at Atletico], but maybe I have a bad couple of years and then they grow to not love me anymore."
Saul has scored once in 10 appearances for Atletico during the 2018-19 campaign.