Champions League winners Chelsea have already submitted a bid to sign the versatile midfielder on a loan deal with an option to buy.
Meanwhile Manchester United have enquired about the player in the last 24 hours but they are yet to come forward with a formal offer.
Saul struggled for regular game time at Atletico Madrid last season and he started just 22 league games for them.
The arrival of Argentine international Rodrigo De Paul will push him further down the pecking order at the Spanish club and he needs to leave in order to play regularly.
Both Chelsea and Manchester United could use more depth in the central midfield and the versatile Spaniard could prove to be a smart addition.
Furthermore signing the player on a loan deal makes it a no-risk signing and the two Premier League clubs should look to wrap up the transfer before the window closes.
Saul will be hoping to regain his confidence and form with regular football in the Premier League this season and it remains to be seen where he ends up eventually.
Both Chelsea and Manchester United could provide him with an excellent platform to showcase his potential.