Sampdoria’s star midfielder has emerged as a prime target for the Gunners this summer.
And Starsport understands a £26.5m bid has been lodged by Arsenal for the Uruguay international.
Torreira is at the World Cup in Russia, so any move will be delayed until after the tournament.
But it appears Arsenal are confident of winning the race for the 22-year-old.
Torreira has a release clause of £22m in his contract, though, meaning the Gunners could be rivalled for his signature.
“Arsenal have shown a willingness to outbid Napoli”
Sampdoria’s Serie A rivals Napoli are keen on luring the player across Italy too.
However, according to The Independent, Arsenal have shown a willingness to outbid Napoli, who are currently the only other serious suitor.
While a deal is not yet fully agreed, the Gunners are in advanced talks with Sampdoria.
Head of recruitment Sven Mislintat is said to be leading the charge.
And if his talks with Sampdoria reach a positive conclusion soon, Torreira will likely be wearing an Arsenal shirt after the World Cup.
The Gunners have put forward a bigger fee than the buyout figure so they can potentially pay in instalments.