Barcelona have been in talks with fellow La Liga club Atletico about a swap deal involving Saul and Antoine Griezmann.
Atletico, who won the Spanish title last season, are prepared to let Saul leave after eight seasons at the club but his next destination is still unclear.
While Barcelona are the frontrunners to secure his signature, AS say he could yet move to the Premier League, with Man United and Liverpool interested in striking a deal.
Saul has been one of the best players in Spain for almost a decade, making 337 appearances for Atletico Madrid and winning La Liga, Copa del Rey and Europa League titles at the club.
Manchester United and Liverpool are both ready to bolster Ole Gunnar Solskjaer and Jurgen Klopp’s respective squads this summer.
The Red Devils are determined to sign a new centre-back partner for club captain Harry Maguire, with Real Madrid’s Raphael Varane their top target.
But Solskjaer is also believed to be open to signing a new midfielder, with the likes of Fred and Scott McTominay flattering to deceive and Paul Pogba regularly linked with a move away from Old Trafford.
Liverpool, meanwhile, may view Saul as an ideal replacement for Georginio Wijnaldum, who has left Anfield to join French giants Paris Saint-Germain.
Klopp’s side only finished third last season following their brilliant title triumph in 2020 and are also expected to sign a new forward to provide competition for Mohamed Salah, Roberto Firmino, Sadio Mane and Diogo Jota.
Manchester United begin their 2021-22 Premier League campaign against Leeds United on August 14, while Liverpool visit Norwich City on the same day.
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