The Gunners are desperate for fresh faces as they look to bounce back from a disappointing campaign.
Unai Emery reportedly has a budget of just £45m but is willing to splash out €30m (£26.9m) on Saliba, who impressed in Ligue 1 last season.
Saint-Etienne will demand the defender back on loan as part of the deal and Arsenal appeared to be in pole position to land the 18-year-old.
But Spurs look to have hijacked the deal in recent days, with reports suggesting representatives will meet today to discuss the deal.
However, according to Yahoo Sport France journalist Manu Lonjon, Wolves have thrown their hat into the ring.
“Tottenham is trying to overtake Arsenal, that’s right,” he tweeted.
“And Wolverhampton is also on the lookout!
“The frame between Arsenal and Sainte was found but negotiations take a lot of time (on bonuses after rmc).
“Wolverhampton is also on the lookout!”
Yahoo Sport France journalist Manu Lonjon
“Tottenham arrived and offered the guaranteed 30M. It is the rule.”
Wolves impressed last season after winning promotion to the Premier League and they stormed to a seventh-place finish.
Nuno Espirito Santo’s side have already splashed out this summer to make Raul Jimenez and Leander Dendoncker’s loan deals permanent.
And they could be set to continue their spending spree if they can convince Saliba to snub north London.
Arsenal find themselves in a defensive crisis, with captain Laurent Koscielny set to leave after refusing to join his team-mates on a pre-season tour of America.
Shkodran Mustafi struggled to perform last season and Emery needs reinforcements before the window closes.
A left-back is also required and Celtic star Kieran Tierney remains their No.1 target.
But Arsenal look to have been priced out of a move for Wilfried Zaha after Crystal Palace demanded £80m.