The 19-year-old spent the 2019-20 campaign on loan at Saint-Etienne, helping the club to reach the Coupe de France final before his temporary contract expired on June 30.
Both Arsenal and Saint-Etienne were keen to extend the deal in order to allow Saliba to play in the final on July 24, but they were unable to agree terms, leading to a public dispute between the two clubs.
However, L'Equipe reports that the Gunners have since performed a U-turn and have held further meetings with representatives of the French outfit which have gone well.
Talks are now thought to be at an advanced stage and Saliba could rejoin Saint-Etienne before the end of this week.
Initial reports had suggested that one more appearance would have resulted in Arsenal having to pay a £2.3m bonus to Saint-Etienne, although it has also been claimed that Arsenal pulled the plug due to the French outfit's refusal to follow their medical demands.
Saint-Etienne face Paris Saint-Germain in the cup final looking to win the trophy for the first time since 1977.