Spurs have launched an offer worth in excess of €27m (£26.5m) plus bonuses for the 18-year-old defender, who was thought to be close to heading to the Emirates Stadium.
Arsenal have an agreement with Saliba but a transfer deal with Saint-Etienne is not yet in place.
A transfer fee has been agreed between the two clubs but the added bonuses are proving to be a sticking point.
According to RMC Sport, Tottenham are set to hold a meeting with the Ligue 1 outfit on Saturday to complete negotiations.
Arsenal must reportedly accelerate their own talks with Saint-Etienne or risk missing out on Saliba.
Saint-Etienne are demanding the France Under-20 international back on loan for the upcoming season as part of any deal.
RMC Sport reporter Mohamed Bouhafsi wrote on Twitter: “Saint Etienne and Arsenal have a global deal on Saliba but could not agree on bonuses.
“Arsenal will have to react because Tottenham and Saint Etienne meet tomorrow to try to finalize the operation because the offer of Tottenham is more important!
“Tottenham have just outbid Arsenal for William Saliba, including €30m excluding bonuses, which goes above & beyond Sainté’s demands.
“ASSE are meeting tomorrow Tottenham to discuss this situation!
“The battle is tough between Arsenal and Tottenham for Saliba.
“Arsenal had an agreement with Sainté and the player.
“Tottenham upsets things with his offer this morning and the meeting tomorrow.
“The next 48 hours are crazy in this issue between the two English clubs.”
Unai Emery had identified Saliba as a long-term option for the Gunners.
The Arsenal boss faces fresh problems at the back after Laurent Koscielny refused to travel to the USA for the club’s pre-season tour.
Saliba made 19 appearances for Saint-Etienne in all competitions last season after breaking into the first team.
He has been tipped for the top thanks to his combination of strength and skill on the ball.