The 26-year-old midfielder wants to leave Atletico Madrid after feeling frustrated at being used in multiple roles in the team.
Atletico Madrid coach Diego Simeone made it clear earlier in the summer that the door is open for the Spaniard to leave the Spanish champions.
Chelsea and Manchester United have emerged as the firm contenders to land Saul in the final week of the transfer window
And it has been claimed that the Blues have already gone in with a bid to take the Spaniard to Stamford Bridge.
Chelsea have offered to sign the midfielder on an initial loan with an option to buy this summer.
Thomas Tuchel wants to bring in a midfield reinforcement after splashing out a club-record fee to sign Romelu Lukaku.
Manchester United are also in the race and have spoken to Atletico Madrid about Saul in the last 24 hours.
But for the moment, Chelsea have the lead in the race to sign Saul in the final week of the transfer window.