Saul has spent his entire senior career at Atletico but is ready for a fresh challenge away from the Spanish capital.
He was also subbed off during crucial games during the season run-in, failing to play the full 90 minutes in their final seven fixtures.
With the midfielder unsettled, The Times report they have now offered him to City in hopes of bringing Bernardo to the Wanda Metropolitano.
The report adds Bernardo is also unsettled at the Etihad, with City said to be keen on signing Jack Grealish to bolster their already impressive options in attacking midfield.
With City also hoping to sign Harry Kane from Tottenham, Atletico are hopeful their player exchange proposal will tempt the Premier League champions from a financial fair play perspective.
Saul has long been linked with a move to Manchester United with Ole Gunnar Solskjaer’s side also still interested in the Spain international, according to recent reports.
The Spaniard has been tipped as one possible replacement for Paul Pogba with the midfielder set to enter the final year of his contract with no new deal agreed.
Chelsea have also enquired into a deal for the Atletico star.
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