The news comes as a blow for Manchester United, who have long coveted the Atleti playmaker.
According to The Times, Atleti are prepared to send Saul in a direct swap for Portugal star Silva.
Reports have claimed Silva has wanted to leave City for over a year and would jump at the chance of playing in Spain.
Having come through the club's academy, Saul is being given the option to decide his next move.
The 26-year-old has played 337 times for his boyhood club.
Atleti chief executive Miguel Angel Gil said: "Saul is captain, he has been with the club for many seasons and therefore has earned the right to decide when and how to end his relationship with Atletico."
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