Griezmann, 30, left Atletico for Barca in 2019 for a fee of €120 million, but the Catalan club are open to letting him leave this summer.
Sources close to Griezmann, who's previously insisted he would like to see out his contract at Camp Nou until 2024, say he would also be willing to return to Atletico if Barca no longer want him.
Barca have reached an agreement in principle for Lionel Messi to extend his contract with the club, but still need to make cutbacks to be able to register the forward's new deal with La Liga.
After Messi, Griezmann was the biggest earner at the club last summer and his departure would create the most room on the wage bill.
Losing Griezmann's salary and replacing it with Saul's would help Barca reduce their outgoings.
However, talks are ongoing as Barca value Griezmann higher than Saul. Therefore, they would like that to be reflected in the deal, be that in terms of Atletico paying an additional fee or including more players.
As well as providing a financial fix, following the signings of Sergio Aguero and Memphis Depay on free transfers this summer, Barca are well-stacked in attack. They also have Ansu Fati to come back from injury.
Saul, meanwhile, is a long-time Barca target who has fallen out of favour under Diego Simeone over the last year.
Having been an ever-present in previous seasons, the 26-year-old started just 22 league games as Atletico won the La Liga title and completed 90 minutes on just 11 occasions.
Barca have previously shown an interest in signing the midfielder and secured a first option on him, which has now expired, when David Villa moved to Atletico in 2013.