A €25 million (£23m/$29m) transfer scramble is expected to be sparked, with the Gunners and Spurs eager to bring a talented 18-year-old onto their books.
Saliba is Unai Emery’s top defensive target at the Emirates Stadium and talks with Saint-Etienne have been opened.
Tottenham, though, are ready to step in and endeavour to pip their arch-rivals to the post if an exit door is opened in France, but it may be that proceedings never get that far, as the Ligue 1 outfit are under no pressure to sell.
Ghislain Printant, the new man in charge at Saint-Etienne, claims that to be the case, with Saliba having indicated that he is prepared to spend at least one more season in his homeland.
"The sale of Saliba is not an absolute necessity," Printant told reporters.
"There is no urgency [to do so]. He is courted by clubs.
"From a sporting point of view we want to keep him next season and that's his wish too. He still has room for development [here].
"We will see. Our goal is for him to be with us again this season."
Saliba enjoyed a breakthrough season with Saint-Etienne in 2018-19, making 17 appearances across all competitions.
Arsenal, meanwhile, conceded a total of 51 goals in the Premier League, with Emery keen to take measures to strengthen that area of his squad.
Shkodran Mustafi has been singled out for criticism for his performances over the last 12 months, while Laurent Koscielny will be 34 in September, with his contract expiring in 2020.
The Gunners have been watching Saliba extensively and have been impressed by his maturity, while his physical attributes fit the mould of the type player that Emery is looking to bring in.
He did commit to a new four-year contract with Saint-Etienne in April, but the Ligue 1 side are aware that they have a fight on their hands in order to get another 12 months out of that agreement.